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BEGIN:VEVENT
SUMMARY:Emma Bailey (CUNY Graduate Center)
DTSTART:20220524T130000Z
DTEND:20220524T132500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/1
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/1/">Large deviations of Selberg's central limit theorem</a>\nby Emma B
 ailey (CUNY Graduate Center) as part of Combinatorial and additive number 
 theory (CANT 2022)\n\n\nAbstract\nSelberg's celebrated central limit theor
 em shows that $\\log\\zeta(1/2+\\rm{i} t)$ at a typical point $t$ at heigh
 t $T$ behaves like a complex\, centered Gaussian random variable with vari
 ance $\\log\\log T$. This talk will present recent results showing that th
 e Gaussian decay persists in the large deviation regime\, at a level on th
 e order of the variance\, improving on the best known bounds in that range
 . Time permitting\, we will also present various applications\, including 
 on the maximum of the zeta function in short intervals. \n\nThis work is j
 oint with Louis-Pierre Arguin.\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jorg Brudern (Universitat Gottingen)
DTSTART:20220524T133000Z
DTEND:20220524T135500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/2
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/2/">Bracketed ternary additive problems</a>\nby Jorg Brudern (Universi
 tat Gottingen) as part of Combinatorial and additive number theory (CANT 2
 022)\n\n\nAbstract\nThe ternary additive problems of Waring's type (that i
 s\, sums of three potentially unlike powers) have attracted many workers i
 n the additive theory of numbers. In this talk\, we discuss several varian
 ts that involve brackets (that is\, the integer part of certain monomials)
 .\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gautami Bhowmik (Universite de Lille)
DTSTART:20220524T143000Z
DTEND:20220524T145500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/4
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/4/">Siegel zeros under Goldbach conjectures</a>\nby Gautami Bhowmik (U
 niversite de Lille) as part of Combinatorial and additive number theory (C
 ANT 2022)\n\n\nAbstract\nA Landau-Siegel zero is a possible though unwelco
 me  counter-example to the  Generalised Riemann Hypothesis. \nProving its 
 absence unconditionally is clearly a difficult problem. We will discuss so
 me results by assuming plausible\nconjectures on the Goldbach problem: the
   Hardy-Litllewood one (1923)\, a weak form due to Fei (2016)\, and a \nwe
 aker form that we studied more recently (Bhowmik-Halupczok\, \nin: Proceed
 ings of CANT 2019 and 2020). Continuing on these lines\,\nFriedlander-Gold
 ston-Iwaniec-Suriajaya (2022) showed that the assumption of Fei's conjectu
 re is enough to disprove the existence of Siegel zeros.\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wijit Yangjit (University of Michigan)
DTSTART:20220527T193000Z
DTEND:20220527T195500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/5
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/5/">On the Montgomery–Vaughan weighted generalization of Hilbert's i
 nequality</a>\nby Wijit Yangjit (University of Michigan) as part of Combin
 atorial and additive number theory (CANT 2022)\n\n\nAbstract\nHilbert's in
 equality states that\n$$\n\\left\\vert\\sum_{m=1}^N\\sum_{n=1\\atop n\\neq
  m}^N\\frac{z_m\\overline{z_n}}{m-n}\\right\\vert\\le C_0\\sum_{n=1}^N\\le
 ft\\vert z_n\\right\\vert^2\,\n$$\nwhere $C_0$ is an absolute constant. In
  1911\, Schur showed that the optimal value of $C_0$ is $\\pi$.\n\nIn 1974
 \, Montgomery and Vaughan proved a weighted generalization of Hilbert's in
 equality and used it to estimate mean values of Dirichlet series. This gen
 eralized Hilbert inequality is important in the theory of the large sieve.
  The optimal constant $C$ in this inequality is known to satisfy $\\pi\\le
  C<\\pi+1$. It is widely conjectured that $C=\\pi$. In this talk\, I will 
 describe the known approaches to obtain an upper bound for $C$\, which pro
 ceed via a special case of a parametric family of inequalities. We analyze
  the optimal constants in this family of inequalities. A corollary is that
  the method in its current form cannot imply an upper bound for $C$ below 
 $3.19$.\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Trevor D. Wooley (Purdue University)
DTSTART:20220524T153000Z
DTEND:20220524T155500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/6
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/6/">Shifted analogues of the divisor function</a>\nby Trevor D. Wooley
  (Purdue University) as part of Combinatorial and additive number theory (
 CANT 2022)\n\n\nAbstract\nSuppose that $\\theta$ is irrational. Then almos
 t all elements \n$\\nu\\in \\mathbb Z[\\theta]$ that may be written as a $
 k$-fold product of the shifted integers \n$n+\\theta$ $(n\\in \\mathbb N)$
  are thus represented essentially uniquely. We discuss this and related pa
 ucity problems. \n\nMost of this work is joint with Winston Heap and Anura
 g Sahay.\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Huy Pham (Stanford University)
DTSTART:20220524T170000Z
DTEND:20220524T172500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/7
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/7/">Homogeneous structures in subset sums and applications</a>\nby Huy
  Pham (Stanford University) as part of Combinatorial and additive number t
 heory (CANT 2022)\n\n\nAbstract\nIn recent joint works with David Conlon a
 nd Jacob Fox\, we develop novel techniques which allow us to prove a diver
 se range of results relating to subset sums. In the one-dimensional case\,
  our techniques imply the existence of long homogeneous arithmetic progres
 sions in the set of subset sums under a variety of assumptions. This allow
 s us to resolve a number of longstanding open problems\, including: soluti
 ons to the three problems of Burr and Erdos on Ramsey complete sequences\,
  for which Erdos later offered a combined total of 350\; analogous results
  for the new notion of density complete sequences\; the solution to a conj
 ecture of Alon and Erdos on the minimum number of colors needed to color t
 he positive integers less than n so that n cannot be written as a monochro
 matic sum\; the exact determination of an extremal function introduced by 
 Erdos and Graham on sets of integers avoiding a given subset sum\; and\, a
 nswering a question reiterated by several authors\, a homogeneous strength
 ening of a result of Szemeredi and Vu on long arithmetic progressions in s
 ubset sums. In follow-up work in the multi-dimensional case\, we show the 
 existence of large homogeneous generalized arithmetic progressions in the 
 set of subset sums of sufficiently large subsets of [n]\, yielding a stren
 gthening of a seminal result of Szemeredi and Vu. As an application\, we m
 ake progress on the Erdos--Straus non-averaging sets problem\, showing tha
 t every subset A of [n] of size at least n^{\\sqrt{2} - 1 + o(1)} contains
  an element which is the average of two or more other elements of A. This 
 gives the first polynomial improvement on a result of Erdos and Sarkozy fr
 om 1990.\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Krystian Gajdzica (Jagiellonian University\, Poland)
DTSTART:20220524T173000Z
DTEND:20220524T175500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/8
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/8/">Some inequalities for the multicolor restricted partition function
  $p_\\mathcal{A}(n\,k)$</a>\nby Krystian Gajdzica (Jagiellonian University
 \, Poland) as part of Combinatorial and additive number theory (CANT 2022)
 \n\n\nAbstract\nFor a non-decreasing sequence of positive integers $\\math
 cal{A}=\\left(a_i\\right)_{i=1}^\\infty$ and a fixed integer $k\\geqslant1
 $\, the multicolor restricted partition function $p_\\mathcal{A}(n\,k)$ co
 unts the number of partitions of $n$ with parts in the multiset $\\{a_1\,a
 _2\,\\ldots\,a_k\\}$. The talk is devoted to some multiplicative inequalit
 ies related to $p_\\mathcal{A}(n\,k)$. Among other things\, we will examin
 e: the Bessenrodt-Ono inequality for $p_\\mathcal{A}(n\,k)$\, the $\\log$-
 concavity of the sequence $\\left(p_\\mathcal{A}(n\,k)\\right)_{n=1}^\\inf
 ty$\, the\nhigher order Tur\\'an property and other similar phenomena.\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:James Sellers (University of Minnesota Duluth)
DTSTART:20220524T180000Z
DTEND:20220524T182500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/9
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/9/">Relating the crank of a partition and smallest missing parts</a>\n
 by James Sellers (University of Minnesota Duluth) as part of Combinatorial
  and additive number theory (CANT 2022)\n\n\nAbstract\nThe primary goal of
  this talk is to demonstrate a natural connection between the smallest mis
 sing part of an integer partition (commonly referred to as the ``mex" of t
 he partition) and the concept of the crank of a partition. After providing
  a brief history of the crank of a partition a la Dyson as well as Andrews
  and Garvan\, we will utilize straightforward generating function manipula
 tions to make this connection. We will then consider additional results on
  the mex statistic based on parity\, and we will also demonstrate connecti
 ons between the crank and Frobenius symbols which satisfy certain conditio
 ns. \n\nThis work is joint with Brian Hopkins\, Dennis Stanton\, and Ae Ja
  Yee.\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Li Guo (Rutgers University - Newark)
DTSTART:20220524T190000Z
DTEND:20220524T192500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/10
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/10/">Renormalization of quasisymmetric functions</a>\nby Li Guo (Rutge
 rs University - Newark) as part of Combinatorial and additive number theor
 y (CANT 2022)\n\n\nAbstract\nThe algebra of quasisymmetric functions (QSym
 ) has played a central role in multiple zeta values and a\nlarge class of 
 combinatorial algebraic structures related to symmetric functions. A natur
 al linear basis of QSym is the set of monomial quasisymmetric functions de
 fined by compositions\, that is\,\nvectors of positive integers.  Extendin
 g such a definition for weak compositions\, that is\, vectors\nof nonnegat
 ive integers\, leads to divergent expressions. This phenomenon is closely 
 related to the divergency of multiple zeta values with nonpositive integer
  arguments. \n\nWe apply\nthe method of renormalization in the spirit of C
 onnes and Kreimer to address  \nthe divergency\, and realize weak composit
 ion\nquasisymmetric functions as power series. \nThe resulting Hopf algebr
 a has the Hopf algebra of\nquasisymmetric functions as both a Hopf subalge
 bra and a Hopf quotient algebra. \n\nThis is joint work with Houyi Yu and 
 Bin Zhang.\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Johann Thiel (New York City College of Technology (CUNY))
DTSTART:20220524T193000Z
DTEND:20220524T195500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/11
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/11/">Solving the membership problem for certain subgroups of $SL_2(\\m
 athbb{Z})$</a>\nby Johann Thiel (New York City College of Technology (CUNY
 )) as part of Combinatorial and additive number theory (CANT 2022)\n\n\nAb
 stract\nFor positive integers $u$ and $v$\, let $L_u=\\begin{bmatrix} 1 & 
 0 \\\\ u & 1 \\end{bmatrix}$ and $R_v=\\begin{bmatrix} 1 & v \\\\ 0 & 1 \\
 end{bmatrix}$. Let $G_{u\,v}$ be the group generated by $L_u$ and $R_v$. T
 he membership problem for $G_{u\,v}$ asks the following question: Given a 
 2-by-2 matrix $M=\\begin{bmatrix}a & b \\\\c & d\\end{bmatrix}$\, is there
  a relatively straightforward method for determining if $M$ is a member of
  $G_{u\,v}$? In the case where $u=2$ and $v=2$\, Sanov was able to show th
 at simply checking some divisibility conditions for $a$\, $b$\, $c$ and $d
 $ is enough to make this determination. We answered this question in the c
 ase where $u\,v\\geq 3$ by finding a characterization of matrices $M$ in $
 G_{u\,v}$ in terms of the short continued fraction representation of $\\fr
 ac{b}{d}$\, extending some results of Esbelin and Gutan. By modifying our 
 previous work\, we are able to further extend our previous result to the m
 ore difficult case where $u\,v\\geq 2$ with $uv\\neq 4$.\n\nThis is joint 
 work with Sandie Han\, Ariane M. Masuda\, and Satyanand Singh.\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chi Hoi Yip (University of British Columbia)
DTSTART:20220524T200000Z
DTEND:20220524T202500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/12
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/12/">Asymptotics for the number of directions determined by $[n] \\tim
 es [n]$ in $\\mathbb{F}_p^2$</a>\nby Chi Hoi Yip (University of British Co
 lumbia) as part of Combinatorial and additive number theory (CANT 2022)\n\
 n\nAbstract\nLet $p$ be a prime and $n$ a positive integer such that $\\sq
 rt{\\frac p2} + 1 \\leq n \\leq \\sqrt{p}$. For any arithmetic progression
  $A$ of length $n$ in $\\mathbb{F}_p$\, we establish an asymptotic formula
  for the number of directions determined by $A \\times A \\subset \\mathbb
 {F}_p^2$. The key idea is to reduce the problem to counting the number of 
 solutions to the bilinear Diophantine equation $ad+bc=p$ in variables $1\\
 le a\,b\,c\,d\\le n$\; our asymptotic formula for the number of solutions 
 is of independent interest. \n\nJoint work with Greg Martin and Ethan Whit
 e.\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ethan Patrick White (University of British Columbia)
DTSTART:20220524T203000Z
DTEND:20220524T205500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/13
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/13/">Erdos' minimum overlap problem</a>\nby Ethan Patrick White (Unive
 rsity of British Columbia) as part of Combinatorial and additive number th
 eory (CANT 2022)\n\n\nAbstract\nIn 1955 Erd\\H{o}s posed the following pro
 blem. Let $n$ be a positive integer and $A\,B \\subset [2n    A weighted g
 eneralization of classical zero-sum constants\nwas introduced by Adhikari 
 {\\it et al.} in 2006 and has been an active area of research since then. 
 In the last fifteen years\, weighted zero-sum constants for $\\mathbb {Z}_
 n$ with several interesting weight sets have been found.\nIn this talk\, w
 e take up the problem of determining the exact values and providing bounds
  of the weighted Davenport constant of $\\mathbb {Z}_n$  \nwith some new w
 eight sets.\n\nNext\, we consider a weighted generalization of the {\\it t
 he Erd\\H{o}s-Ginzburg-Ziv constant}. \nLet $G$ be a  finite abelian group
  with $\\exp(G)=n$. For a positive integer $k$ and a non-empty subset $A$ 
 of $[1\, n-1]$\,\nthe arithmetical invariant $\\mathsf s_{kn\,A}(G)$  is d
 efined to be the  least positive integer $t$ such that\nany sequence $S$ o
 f $t$ elements in $G$ has an $A$-{\\it weighted zero-sum subsequence} of  
 length $kn$.\nWe give the exact value of $\\mathsf s_{kq\,A}(G)$\, for int
 egers $k\\geq 2$ and $A=\\{1\,2\\}$\,\nwhere $G$ is an abelian $p$-group w
 ith $rank(G)\\leq 4$\, $p$ is an odd prime  and $exp(G)=q$.\nOur method co
 nsists of a modification of a polynomial method \nof R\\'onyai.\n\nLastly\
 , we consider the questions regarding inverse problems for the weighted ze
 ro-sum constants of $\\mathbb {Z}_n$. An inverse problem is the problem of
  characterizing all the weighted {\\it zero-sum free sequences} over $\\ma
 thbb {Z}_n$ of specific lengths for the particular weight sets under consi
 deration.\n\nThis work was joint with Sukumar Das Adhikari and partly with
  Md Ibrahim Molla and Subha Sarkar.\n]$ be a partition of $[2n]$ such that
  $|A|=|B| = n$. For any such partition and integer $-2n<k<2n$\, define $M_
 k$ to be the number of solutions $(a\,b) \\in A  \\times B$ to $a-b = k$. 
 Estimate the size of the function\n\\[ M(n) = \\min_{A\\cup B = [2n]} \\ma
 x_{-2n<k<2n} M_k\,\\]\nwhere the minimum is taken over all partitions of $
 [2n]$ into equal-sized sets. Many upper and lower estimates were obtained 
 over the following decades\, and the state of the art is $0.356 < M(n)/n <
  0.381$. We use elementary Fourier analysis to translate the problem to a 
 convex optimization program and obtain the new lower bound $M(n)/n>0.379$.
 \n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Benjamin Baily (Williams College)
DTSTART:20220524T183000Z
DTEND:20220524T185500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/14
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/14/">Large sets are sumsets</a>\nby Benjamin Baily (Williams College) 
 as part of Combinatorial and additive number theory (CANT 2022)\n\n\nAbstr
 act\nLet $[n] :=\\{0\,1\,2\,\\dots\,n\\}$. Intuitively\, all large subsets
  of $[n]$ have additive structure\, and Roth famously made this precise by
 \nfinding constants $c$\, $N > 0$ such that for $n \\geq N$\, any subset o
 f $[n]$ containing more than $\\frac{cn}{\\log\\log n}$ elements must cont
 ain an\narithmetic progression of length $3$. We establish a different int
 erpretation of the intuition by finding explicit constants $\\alpha = \\fr
 ac{1}{\\log\n2}$ and $\\beta = \\frac{1}{\\log 1.325}$ such that\, for suf
 ficiently large $n$\, we have:\n\\begin{enumerate}\n\\item[(i)] any subset
  of $[n]$ with more than $n-\\alpha \\log n$ elements has a nontrivial dec
 omposition as the sum of two sets\, and\n\n\\item [(ii)]there exists a sub
 set of $[n]$ of size $n - \\beta \\log n$ at least  that has no such decom
 position.\n\n\\end{enumerate} We also prove\, using these methods\, a high
 er-dimensional\nanalogue of results (i) and (ii). Notably\, our threshold 
 at which\nstructure appears is far higher than Roth's.\n\nThis work was jo
 int with Justine Dell\, Sophia Dever\, Adam Dionne\, Faye\nJackson\, Leo G
 oldmakher\, Gal Gross\, Steven J. Miller\, Ethan Pesikoff\, Huy\nTuan Pham
 \, Luke Reifenberg\, and Vidya Venkatesh.     \\\\\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shruti S Hegde (Ramakrishna Mission Vivekananda Educational and Re
 search Institute\, India)
DTSTART:20220525T130000Z
DTEND:20220525T132500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/15
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/15/">Weighted zero-sum constants and inverse results</a>\nby Shruti S 
 Hegde (Ramakrishna Mission Vivekananda Educational and Research Institute\
 , India) as part of Combinatorial and additive number theory (CANT 2022)\n
 \n\nAbstract\nA weighted generalization of classical zero-sum constants\nw
 as introduced by Adhikari et al. in 2006 and has been an active area of re
 search since then. In the last fifteen years\, weighted zero-sum constants
  for $\\mathbb {Z}_n$ with several interesting weight sets have been found
 .\nIn this talk\, we take up the problem of determining the exact values a
 nd providing bounds of weighted Davenport constant of $\\mathbb {Z}_n$  \n
 with some new weight sets.\n\nNext\, we consider a weighted generalization
  of the Erd\\H{o}s-Ginzburg-Ziv constant. \nLet $G$ be a  finite abelian g
 roup with $\\exp(G)=n$. For a positive integer $k$ and a non-empty subset 
 $A$ of $[1\, n-1]$\,\nthe arithmetical invariant $\\mathsf s_{kn\,A}(G)$  
 is defined to be the  least positive integer $t$ such that\nany sequence o
 f $t$ elements in $G$ has an $A$- weighted zero-sum subsequence of  length
  $kn$.\nWe give the exact value of $\\mathsf s_{kq\,A}(G)$\, for integers 
 $k\\geq 2$ and $A=\\{1\,2\\}$\,\nwhen $G$ is an abelian $p$-group with $ra
 nk(G)\\leq 4$\, $p$ is an odd prime  and $exp(G)=q$.\nOur method consists 
 of a modification of a polynomial method \nof R\\'onyai.\n\nLastly\, we co
 nsider the questions regarding inverse problems for the weighted zero-sum 
 constants of $\\mathbb {Z}_n$. An inverse problem is a problem of characte
 rizing all the weighted {\\it zero-sum free sequences} over $\\mathbb {Z}_
 n$ of specific lengths for the particular weight sets under consideration.
 \n\nThis work was joint with Sukumar Das Adhikari and partly with Md Ibrah
 im Molla and Subha Sarkar.\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gabor Somlai (Eotvos Lorand University and   Alfred Renyi Institut
 e of Mathematics)
DTSTART:20220525T133000Z
DTEND:20220525T135500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/17
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/17/">Fuglede's conjecture\, the one dimensional case</a>\nby Gabor Som
 lai (Eotvos Lorand University and   Alfred Renyi Institute of Mathematics)
  as part of Combinatorial and additive number theory (CANT 2022)\n\n\nAbst
 ract\nFuglede conjectured that a bounded measurable set (in $\\mathbb{R}^n
 $) is spectral if and only if it is a tile. The conjecture was also confir
 med by Fuglede for sets whose tiling complement is lattice and for spectra
 l sets one of whose spectrums is a lattice. \nThe conjecture was disproved
  by Tao by constructing a spectral set in $\\mathbb{Z}_3^5$\, which is not
  a tile and lifted it to the $5$ dimensional Euclidean space. \n\nThe conj
 ecture is open only in dimensions 1 and 2. The 1 dimensional case is direc
 tly connected with the one of finite cyclic groups and to the so called Co
 ven-Meyerowitz conjecture. One of the main aims of the talk is to present 
 some of the methods developed that lead to our recent results.\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Leonid Fel (Technion -- Israel Institute of Technology\, Israel)
DTSTART:20220525T140000Z
DTEND:20220525T142500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/18
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/18/">Commutative monoid of self-dual symmetric polynomials</a>\nby Leo
 nid Fel (Technion -- Israel Institute of Technology\, Israel) as part of C
 ombinatorial and additive number theory (CANT 2022)\n\n\nAbstract\nWe cons
 ider a set ${\\mathfrak R}{\\mathfrak S}\\left(\\lambda\,S_n\\right)$ of s
 elf-\nand skew-reciprocal polynomials in $\\lambda$\, of degree $mn$\, whe
 re $m\\in{\n\\mathbb Z}_{\\geq}$\, $n\\in{\\mathbb Z}_>$\, based on polyno
 mial invariants $I_{n\,\nr}({\\bf x}^n)$ of symmetric group $S_n$\, acting
  on the Euclidean space ${\\mathbb\nE}^n$ over the field of real numbers $
 {\\mathbb R}$\, where ${\\bf x^n}=\\{x_1\,\n\\ldots\,x_n\\}\\in{\\mathbb E
 }^n$. We prove that ${\\mathfrak R}{\\mathfrak S}\\left(\n\\lambda\,S_n\\r
 ight)$ exhibits a commutative monoid under multiplication. Real\nsolutions
  $\\lambda\\left({\\bf x^n}\\right)$ of skew-reciprocal equations have\nma
 ny remarkable properties: a homogeneity of the 1st order\, a duality under
 \ninversion of variables $x_i\\to x_i^{-1}$ and function $\\lambda\\to\\la
 mbda^{-1}$\,\na monotony of $\\lambda\\left({\\bf x^n}\\right)$ with respe
 ct to every $x_i$ and\nothers. We find the bounds of $\\lambda\\left({\\bf
  x^n}\\right)$ which are given \nby arithmetic and harmonic means of the s
 et $\\{x_1\,\\ldots\,x_n\\}$.\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jakub Konieczny (Claude Bernard University Lyon 1\, France)
DTSTART:20220525T143000Z
DTEND:20220525T145500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/19
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/19/">Automatic semigroups</a>\nby Jakub Konieczny (Claude Bernard Univ
 ersity Lyon 1\, France) as part of Combinatorial and additive number theor
 y (CANT 2022)\n\n\nAbstract\nAutomatic sequences\, that is\, sequences com
 putable by finite automata\, have been extensively studied from a variety 
 of perspectives\, including combinatorics\, number theory\, dynamics and t
 heoretical computer science. Classification problems are a natural class o
 f questions in the theory of automatic sequences. In particular\, the prob
 lem of classifying automatic multiplicative sequences has attracted consid
 erable attention\, culminating in complete classification which we obtaine
 d in joint work with Clemens M\\"{u}llner and Mariusz Lema\\'{n}czyk. The 
 subject of my talk will be an extension of this line of inquiry\, which we
  pursue in joint work with Oleksiy Klurman. Under mild technical assumptio
 ns\, we classify all automatic multiplicative semigroups\, that is\, all s
 ets $E$ of integers which are closed under multiplication and such that th
 e indicator function $1_E$ is automatic. Additionally\, we show (again\, u
 nder mild technical assumptions) that if $E\,F$ are automatic sets with $E
  \\cdot F \\subset E$ then $E$   must contain a large essentially periodic
  component. This leads to potentially interesting open problems concerning
  products of automatic sets.\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Pal Pach (TU Budapest)
DTSTART:20220525T150000Z
DTEND:20220525T152500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/20
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/20/">Colouring the smooth numbers</a>\nby Peter Pal Pach (TU Budapest)
  as part of Combinatorial and additive number theory (CANT 2022)\n\n\nAbst
 ract\nFor a given $n$\, can we colour the positive integers using precisel
 y $n$ colours in such a way that for any $a$\, the numbers $a\, 2a\, \\dot
 s\, na$ all get different colours? This question is still open in general.
  I will present a survey of known results and some other problems it leads
  to. \n\nThis is joint work with Andros Caicedo and Thomas Chartier.\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jared Duker Lichtman (University of Oxford)
DTSTART:20220525T153000Z
DTEND:20220525T155500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/21
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/21/">A proof of the Erdos primitive set conjecture</a>\nby Jared Duker
  Lichtman (University of Oxford) as part of Combinatorial and additive num
 ber theory (CANT 2022)\n\n\nAbstract\nA set of integers greater than 1 is 
 primitive if no member in the set divides another. Erdos proved in 1935 th
 at the series of $1/(n\\log n)$\, ranging over $n$ in $A$\, is uniformly b
 ounded over all choices of primitive sets $A$. In 1988 he asked if this bo
 und is attained for the set of prime numbers. In this talk we describe rec
 ent work which answers Erdos' conjecture in the affirmative. We will also 
 discuss applications to old questions of Erdos\, Sarkozy\, and Szemeredi f
 rom the 1960s.\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Max Wenqiang Xu (Stanford University)
DTSTART:20220525T170000Z
DTEND:20220525T172500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/22
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/22/">On a Turan conjecture and random multiplicative functions</a>\nby
  Max Wenqiang Xu (Stanford University) as part of Combinatorial and additi
 ve number theory (CANT 2022)\n\n\nAbstract\nWe show that if $f$ is the ran
 dom completely multiplicative function\, \nthe probability that $\\sum_{n\
 \le x}\\frac{f(n)}{n}$ is positive for every $x$ is at least \\\\\n$1-10^{
 -40}$. For large $x$  we prove an asymptotic upper bound of \\\\\n$O(\\exp
 (-\\exp( \\frac{\\log x}{C\\log \\log x })))$ on the probability that a pa
 rticular truncation is negative. 	\nThis is joint work with Rodrigo Angelo
 .\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Piotr Miska (Jagiellonian University\, Krakow\, Poland)
DTSTART:20220525T173000Z
DTEND:20220525T175500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/23
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/23/">On (non-)realizibility of Stirling numbers</a>\nby Piotr Miska (J
 agiellonian University\, Krakow\, Poland) as part of Combinatorial and add
 itive number theory (CANT 2022)\n\n\nAbstract\nWe say that a sequence $(a_
 n)_{n\\in\\mathbb{N}_+}$ of non-negative integers is realizable if there e
 xists a set $X$ and a mapping $T : X \\to X$ such that $a_n$ is the number
  of fixed points of $T^n$. For each $k \\in\\mathbb{N}_+$ and $j \\in \\{1
 \,2\\}$ we define a sequence $S^{(j)}_k =(S^{(j)}(n+k -1\,k))_{n\\in\\math
 bb{N}_+}$ \, where $S^{(j)}(n\,k)$ is the Stirling number of the $j$-th ki
 nd (in case of $j = 1$ we consider unsigned Stirling numbers). The aim of 
 the talk is to prove that $S^{(2)}_k$ is realizable if and only if $k \\in
  \\{1\,2\\}$\, while for $k \\geq 3$ the sequence $S^{(2)}_k$ is almost re
 alizable with a failure $(k-1)!$\, i. e. $(k-1)!S^{(2)}_k$ is realizable. 
 Moreover\, I will show that for each $k \\in\\mathbb{N}_+$ the sequence $S
 ^{(1)}_k$ is not almost realizable\, i. e. for any $r \\in\\mathbb{N}_+$ t
 he sequence $rS^{(1)}_k$ is not realizable. \n\nThe talk is based on a joi
 nt work with Tom Ward (Newcastle\, UK).\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Qinghai Zhong (University of Graz\, Austria)
DTSTART:20220525T180000Z
DTEND:20220525T182500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/24
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/24/">On monoids of  weighted zero-sum sequences</a>\nby Qinghai Zhong 
 (University of Graz\, Austria) as part of Combinatorial and additive numbe
 r theory (CANT 2022)\n\n\nAbstract\nLet $G$ be an additive finite abelian 
 group and $\\Gamma \\subset \\operatorname{End} (G)$ be a subset of the en
 domorphism group of $G$. A sequence $S = g_1 \\cdot \\ldots \\cdot g_{\\el
 l}$ over $G$ is a ($\\Gamma$-)weighted zero-sum sequence if there are $\\g
 amma_1\, \\ldots\, \\gamma_{\\ell} \\in \\Gamma$ such that $\\gamma_1 (g_1
 ) + \\ldots + \\gamma_{\\ell} (g_{\\ell})=0$.  We study  algebraic and ari
 thmetic properties of  monoids of weighted zero-sum sequences.\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sinai Robins (University of Sao Paulo\, Brazil)
DTSTART:20220525T183000Z
DTEND:20220525T185500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/25
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/25/">The covariogram and an extension of Siegel's formula</a>\nby Sina
 i Robins (University of Sao Paulo\, Brazil) as part of Combinatorial and a
 dditive number theory (CANT 2022)\n\n\nAbstract\nWe extend a formula of Ca
 rl Ludwig Siegel in the geometry of numbers.\nSiegel's original formula as
 sumed that there is exactly one lattice point in the interior of the body\
 , while here\nwe relax that condition\, so that the body may contain an ar
 bitrary number of interior lattice points.  Our extension involves a latti
 ce sum of the covariogram for any compact set  $\\mathcal K \\subset \\mat
 hbb{R}^d$\,  where the covariogram of $\\mathcal K$ at  $x \\in \\mathbb R
 ^d$ is defined by $\\rm{vol}$$( \\mathcal K \\cap (\\mathcal K + x))$.   \
 nThe proof hinges on a variation of the Poisson summation formula which we
  derive here\, and the Fourier methods herein also allow for more general 
 admissible sets.  One of the consequences of these results is a new charac
 terization of multi-tilings of Euclidean space by translations\, using the
  lower bound on lattice sums of such covariograms.  The classical result k
 nown as Van der Corput's inequality\, also follows immediately from the ma
 in result\, as well as a new spectral formula for the volume of a compact 
 set.  \n\nThis is joint work with Michel Faleiros Martins.\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (Lehman College (CUNY))
DTSTART:20220525T190000Z
DTEND:20220525T192500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/26
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/26/">Multiplicity interpolation of polynomials</a>\nby Mel Nathanson (
 Lehman College (CUNY)) as part of Combinatorial and additive number theory
  (CANT 2022)\n\n\nAbstract\nInterpolation problems related to the theorems
  of Descartes\, Budan-Fourier\, and Sturm in the theory of equations.\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Noah Kravitz (Princeton University)
DTSTART:20220525T193000Z
DTEND:20220525T195500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/27
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/27/">Zero patterns of derivatives of polynomials</a>\nby Noah Kravitz 
 (Princeton University) as part of Combinatorial and additive number theory
  (CANT 2022)\n\n\nAbstract\nMotivated by recent work of Nathanson\, we stu
 dy the zero patterns of derivatives of polynomials.  For $P$ a polynomial 
 of degree $n$ and $\\Lambda=(\\lambda_1\, \\ldots\, \\lambda_m)$ an $m$-tu
 ple of distinct complex numbers\, we consider the $m \\times (n+1)$ \\emph
 {dope matrix} $D_P(\\Lambda)$ whose $ij$-entry equals $1$ if $P^{(j)}(\\la
 mbda_i)=0$ and equals $0$ otherwise (for $1 \\leq i \\leq m$\, $0 \\leq j 
 \\leq n$).  We address several natural questions: When $m$ is $1$ or $2$\,
  what do the possible dope matrices look like\, and how many are there?  W
 hat can we say about general upper bounds on the number of $m \\times (n+1
 )$ dope matrices?  For which $m$-tuples $\\Lambda$ is the number of $m \\t
 imes (n+1)$ dope matrices maximized?  Does every $\\{0\,1\\}$-matrix appea
 r as the left-most portion of some dope matrix?  \n\nBased on joint work w
 ith Noga Alon and Kevin O'Bryant.\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Catherine Yan (Texas A&M University)
DTSTART:20220525T200000Z
DTEND:20220525T202500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/28
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/28/">Multivariate  Goncarov polynomials and integer sequences</a>\nby 
 Catherine Yan (Texas A&M University) as part of Combinatorial and additive
  number theory (CANT 2022)\n\n\nAbstract\nUnivariate delta Gon\\v{c}arov p
 olynomials arise when the classical Gon\\v{c}arov interpolation problem in
  numerical analysis is modified by replacing derivatives with delta operat
 ors. When the delta operator under consideration is the backward differenc
 e operator\, we acquire the univariate difference Gon\\v{c}arov polynomial
 s\, which have a combinatorial relation to lattice paths in the plane with
  a given right boundary. In this talk\, we extend several algebraic and an
 alytic properties of univariate  Gon\\v{c}arov polynomials to the multivar
 iate case with both the derivative and backward difference operators. We t
 hen establish a combinatorial interpretation of multivariate  Gon\\v{c}aro
 v polynomials in terms of certain constraints on $d$-tuples of integer seq
 uences. This motivates a connection between multivariate  Gon\\v{c}arov po
 lynomials and a higher-dimensional generalized parking function\, the $\\m
 athbf{U}$-parking function\, from which we derive several enumerative resu
 lts based on the theory of delta operators.   \n\nThis talk is based on jo
 int work with Ayo Adeniran and Lauren Snider.\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yin Choi Cheng (CUNY Graduate Center)
DTSTART:20220525T203000Z
DTEND:20220525T205500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/29
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/29/">Order type of shifts of morphic words</a>\nby Yin Choi Cheng (CUN
 Y Graduate Center) as part of Combinatorial and additive number theory (CA
 NT 2022)\n\n\nAbstract\nThe shifts of an infinite word $W=a_0a_1\\cdots$ a
 re the words $W_i=a_ia_{i+1}\\cdots$. As a measure of the complexity of a 
 word $W$\, we consider the order-type of the set of shifts\, ordered lexic
 ographically. We will look at the order-type of shifts of morphic words ov
 er a finite alphabet that are not ultimately periodic. As a concrete examp
 le\, we give the explicit ordering among shifts of the Thue-Morse word. Th
 e order type of shifts of the Fibonacci word will be discussed. We then gi
 ve special consideration to uniform morphisms on 3 letters.\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tim Trudgian (UNSW Canberra at the Australian Defence Force Academ
 y)
DTSTART:20220525T210000Z
DTEND:20220525T212500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/30
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/30/">Don’t believe the Fake Mu’s!</a>\nby Tim Trudgian (UNSW Canbe
 rra at the Australian Defence Force Academy) as part of Combinatorial and 
 additive number theory (CANT 2022)\n\n\nAbstract\nPerhaps your favourite s
 um is biased \\ldots leaning a little towards the negative\, perhaps? Perh
 aps your sum is suspiciously similar to the Moebius function $\\mu(n)$? Wh
 at can we do with such fake mu’s? Come along to find out\, and together\
 , we can make arithmetic great again!\n\nThis is joint work with Greg Mart
 in (UBC) and Mike Mossinghoff (CCR\, Princeton).\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jin-Hui Fang (Nanjing University of Information Science and Techno
 logy)
DTSTART:20220526T130000Z
DTEND:20220526T132500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/31
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/31/">Representation functions avoiding integers with density zero</a>\
 nby Jin-Hui Fang (Nanjing University of Information Science and Technology
 ) as part of Combinatorial and additive number theory (CANT 2022)\n\n\nAbs
 tract\nFor a nonempty set $A$ of integers and any integer $n$\, denote $r_
 {A}(n)$ by the number of representations of $n$ of the form $n=a+a'$\, whe
 re $a\\le a'$ and $a\,a'\\in A$ and $d_{A}(n)$ by the number of pairs $(a\
 ,a')$ with $a\,a'\\in A$ such that $n=a-a'$. In 2008\, Nathanson considere
 d the representation function with infinitely many zeros. Following Nathan
 son's work\, we proved that\, for any set $T$ of integers with density zer
 o\, there exists a sequence $A$ of integers such that $r_A(n)=1$ for all i
 ntegers $n\\not\\in T$ and $r_A(n)=0$ for all integers $n\\in T$\, and $d_
 A(n)=1$ for all positive integers $n$. We will also present our recent res
 ults on representation functions.\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carlo Sanna (Politecnico di Torino\, Italy)
DTSTART:20220526T133000Z
DTEND:20220526T135500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/32
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/32/">Membership in random ratio sets</a>\nby Carlo Sanna (Politecnico 
 di Torino\, Italy) as part of Combinatorial and additive number theory (CA
 NT 2022)\n\n\nAbstract\nLet $\\mathcal{A}$ be a random set constructed by 
 picking independently each element of $\\{1\, \\dots\, n\\}$ with probabil
 ity $\\alpha \\in (0\, 1)$.\nSeveral authors studied combinatorial/number-
 theoretic objects involving $\\mathcal{A}$\, including the sum set $\\math
 cal{A} + \\mathcal{A}$\, the product set $\\mathcal{A}\\mathcal{A}$\, and 
 the ratio set $\\mathcal{A} /\\! \\mathcal{A}$.\nGeneralizing a previous r
 esult of Cilleruelo and Guijarro-Ord\\'{o}\\~{n}ez\, we give a formula for
  the probability that a rational number $q$ belongs to the ratio set $\\ma
 thcal{A} /\\! \\mathcal{A}$.\nMoreover\, we give some results about formul
 as for the probability of the event $\\bigvee_{i=1}^k\\!\\big(q_i \\in \\m
 athcal{A} /\\! \\mathcal{A}\\big)$\, where $q_1\, \\dots\, q_k$ are ration
 al numbers\, showing that they are related to the study of the connected c
 omponents of certain graphs.\nFinally\, we provide some open question for 
 future research.\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Norbert Hegyvari (Eotvos Lorand University and   Alfred Renyi Inst
 itute of Mathematics\, Hungary)
DTSTART:20220526T140000Z
DTEND:20220526T142500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/33
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/33/">Boolean functions defined on pseudo-recursive sequences</a>\nby N
 orbert Hegyvari (Eotvos Lorand University and   Alfred Renyi Institute of 
 Mathematics\, Hungary) as part of Combinatorial and additive number theory
  (CANT 2022)\n\n\nAbstract\nWe define Boolean functions on hypergraphs wit
 h edges having big intersections\, and an opposite situation\, \nhypergrap
 hs which are thinly intersective induced by pseudo-recursive sequences. As
  a main result\, we estimate the cardinality of their supports.\nA sequenc
 e $X$ is said to be pseudo-recursive (or pesudo-linear) sequence if the id
 entity\n$x_{n+1}=M\\cdot x_n+ b_{j_{n+1}}$ holds\, where $ b_{j_{n+1}}\\in
  \\{b_1\,b_2\, \\dots b_k\\}$) for $n \\geq 0$ and $M$ is a positive integ
 er. (This type of sequences have a long list in the combinatorial number t
 heory and other areas too\, e.g. in random walk theory). \n\n The tools co
 me from additive combinatorics and the uncertainty inequality.\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Steven Senger (Missouri State University)
DTSTART:20220526T143000Z
DTEND:20220526T145500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/34
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/34/">Distinct dot products\, convexity\, and AA+1</a>\nby Steven Senge
 r (Missouri State University) as part of Combinatorial and additive number
  theory (CANT 2022)\n\n\nAbstract\nWe discuss recent developments in estim
 ating the number of distinct dot products determined by a large finite set
  of $n$ points in the plane. The improvement comes from improved understan
 ding of the multiplicative structure of an additively shifted product set\
 , $AA+1\,$ when $A$ is a large finite subset of the real numbers. This bre
 akthrough was made possible by new additive combinatorial results about co
 nvex sets of numbers.\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gergely Kiss (Alfred Renyi Institute of Mathematics\, Hungary)
DTSTART:20220526T150000Z
DTEND:20220526T152500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/35
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/35/">Fuglede's conjecture on the direct product of finite abelian grou
 ps</a>\nby Gergely Kiss (Alfred Renyi Institute of Mathematics\, Hungary) 
 as part of Combinatorial and additive number theory (CANT 2022)\n\n\nAbstr
 act\nWe investigate Fuglede's conjecture on the direct product of abelian 
 groups and its connection to the conjecture in $\\mathbb{R}^n$ for $n\\ge 
 2$. We overview the earlier results: Some important constructions will be 
 shown\, which disproves the conjecture in higher dimensions\, and some tec
 hniques and ideas will be presented\, which serves to prove the conjecture
  for certain abelian groups. Finally we will discuss some developments of 
 the most recent directions of research. This talk is closely related to th
 e talk of Gábor Somlai's about Fuglede's conjecture in the cyclic group a
 nd the one dimensional cases.\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Renling Jin (College of Charleston)
DTSTART:20220526T153000Z
DTEND:20220526T155500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/36
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/36/">Hyper-hyper-hyper-integers</a>\nby Renling Jin (College of Charle
 ston) as part of Combinatorial and additive number theory (CANT 2022)\n\n\
 nAbstract\nIn a conference five years ago\, T. Tao reported \nhis effort t
 o simplify Szemer\\'{e}di's original combinatorial proof of \nSzemer\\'{e}
 di's theorem using nonstandard analysis. \nWe continued his effort and pre
 sented a simple proof of\nthe theorem for $k=4$ in CANT 2020. In this talk
 \, we will present \na simple proof of the theorem for all $k$. One of the
  main simplifications\nis that a Tower of Hanoi type induction used by Sze
 mer\\'{e}di as well as Tao\nis replaced by a straightforward induction. In
  the proof the integers with\nthree levels of infinities are used.\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rachel Greenfeld (UCLA)
DTSTART:20220526T170000Z
DTEND:20220526T172500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/37
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/37/">Translational tilings</a>\nby Rachel Greenfeld (UCLA) as part of 
 Combinatorial and additive number theory (CANT 2022)\n\n\nAbstract\nTransl
 ational tiling is a covering of a space using translated copies of some bu
 ilding blocks\, called the "tiles" without any positive measure overlaps. 
 Which are the possible ways that a space can be tiled? In the talk\, we wi
 ll discuss the study of this question as well as its applications\, and re
 port on recent progress\, joint with Terence Tao.\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Thai Hoang Le (University of Mississippi)
DTSTART:20220526T173000Z
DTEND:20220526T175500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/38
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/38/">Bohr sets in sumsets in countable abelian groups</a>\nby Thai Hoa
 ng Le (University of Mississippi) as part of Combinatorial and additive nu
 mber theory (CANT 2022)\n\n\nAbstract\nA \\textit{Bohr set} in an abelian 
 topological group $G$ is a subset of the form\n\\[\nB(K\, \\epsilon) = \\{
  g \\in G: |\\chi(g) - 1| < \\epsilon \\\, \\forall \\chi \\in K \\}\n\\]\
 nwhere $K$ is a finite subset of the dual group $\\widehat{G}$. A classica
 l theorem of Bogolyubov says that if $A \\subset \\mathbf{Z}$ has positive
  upper density $\\delta$\, then $A+A-A-A$ contains a Bohr set $B(K\, \\eps
 ilon)$ where $|K|$ and $\\epsilon$ depend only on $\\delta$. While the sam
 e statement for $A-A$ is not true (a result of K\\v{r}\\'i\\v{z})\, Bergel
 son and Ruzsa proved that if $r+s+t=0$\, then $rA + sA+tA$ contains a Bohr
  set (here $rA = \\{ ra: a \\in A \\}$). \nWe   investigate this phenomeno
 n in more general groups $G$\, where $rA\, sA\, tA$ are replaced by images
  of $A$ under certain endomomorphisms of $G$. It is also natural to ask fo
 r partition analogues of the Bergelson-Ruzsa theorem. In CANT 2021\, I dis
 cussed our results in compact abelian groups (generalizations of $\\mathbf
 {R} /\\mathbf{Z}$). \\\nIn this talk\, I will discuss our progress on coun
 table discrete abelian groups (generalizations of $\\mathbf{Z}$). The key 
 ingredients are certain transference principles which allow us to transfer
  the results from compact groups to discrete countable groups. \nThis talk
  is based on joint works with Anh Le\, and with Anh Le and John Griesmer.\
 n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Paul Pollack (University of Georgia)
DTSTART:20220526T180000Z
DTEND:20220526T182500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/39
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/39/">Weak uniform distribution of certain arithmetic functions</a>\nby
  Paul Pollack (University of Georgia) as part of Combinatorial and additiv
 e number theory (CANT 2022)\n\n\nAbstract\nFor any fixed integer $q$\, it 
 is a classical result (implicit in work of Landau\, and perhaps known earl
 ier) that Euler's function $\\phi(n)$ is a multiple of $q$ asymptotically 
 100\\% of the time. Thus\, $\\phi(n)$ is very far from being uniformly dis
 tributed mod $q$ in the usual sense (unless $q=1$ !). On the other hand\, 
 Narkiewicz has proved that $\\phi(n)$ is weakly uniformly distributed mod 
 $q$ whenever $q$ is coprime to 6\; “weakly” means that every coprime r
 esidue class mod $q$ gets its fair share of values $\\phi(n)$\, from among
  the $n$ with $\\phi(n)$ coprime to $q$. In fact\, Narkiewicz proves this 
 not just for $\\phi$ but for a wide class of “polynomially-defined” mu
 ltiplicative functions. In this talk\, we will consider these weak uniform
  distribution problems with an eye towards obtaining wide ranges of unifor
 mity in the modulus $q$. \n\nThis is joint work with Noah Lebowitz-Lockard
  and Akash Singha Roy.\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Junxuan Shen (California Institute of Technology)
DTSTART:20220526T183000Z
DTEND:20220526T185500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/40
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/40/">The structural incidence problem for cartesian products</a>\nby J
 unxuan Shen (California Institute of Technology) as part of Combinatorial 
 and additive number theory (CANT 2022)\n\n\nAbstract\nWe prove new structu
 ral results for point-line incidences. An incidence is a pair of one point
  and one line\, where the point is on the line. The Szemer\\'{e}di-Trotter
  theorem states that $n$ points and $n$ lines form $O(n^{4/3})$ incidences
 . This bound has been used to obtain many results in combinatorics\, numbe
 r theory\, harmonic analysis\, and more. While the Szemer\\'{e}di-Trotter 
 bound has been known for several decades\, the structural problem remains 
 wide-open. This problem asks to characterize the point-line configurations
  with $\\Theta(n^{4/3})$ incidences. \nWe prove that when the point set $\
 \mathcal{P}$ is a Cartesian product where only one axis of it behaves like
  a lattice\, the line set must contain many families of parallel lines to 
 achieve the maximal incidence bound.\n\nTheorem: \nConsider $1/3<\\alpha<2
 /3$. Let $A\,B\\subset\\RR$ satisfy that $A=\\{1\,2\,\\cdots\, n^{\\alpha}
 \\}$ and $|B|=n^{1-\\alpha}$. Let $\\mathcal{L}$ be a set of $n$ lines in 
 $\\RR^2$\, such that $I(A\\times B\,\\mathcal{L})=\\Theta(n^{4/3})$. Then 
 $\\mathcal{L}$ contains $\\Omega(n^{1-\\beta}/\\log n)$ disjoint families 
 of $\\Theta(n^{\\beta})$ parallel lines for $1-2\\alpha\\le\\beta\\le 2/3$
 .\n\nWhen $\\alpha<1/3$ or $\\alpha>2/3$\, it is impossible to have $\\The
 ta(n^{4/3})$ incidences. We also completely characterize the line set when
  the point set is a lattice.\n\nJoint work with Adam Sheffer. \\\\\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Henry Fleischmann (University of Michigan)
DTSTART:20220526T190000Z
DTEND:20220526T192500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/41
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/41/">Angle variants of the Erd\\H{o}s distinct distance problem</a>\nb
 y Henry Fleischmann (University of Michigan) as part of Combinatorial and 
 additive number theory (CANT 2022)\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Rice (Millsaps College)
DTSTART:20220526T193000Z
DTEND:20220526T195500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/42
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/42/">New results in classical and arithmetic Ramsey theory</a>\nby Ale
 x Rice (Millsaps College) as part of Combinatorial and additive number the
 ory (CANT 2022)\n\n\nAbstract\nFor $r\,k\\in \n$\, Ramsey's Theorem says t
 hat there exists a least positive integer $R_r(k)$ such that every $r$-col
 oring of the edges of a complete graph on $N\\geq R_r(k)$ vertices yields 
 a monochromatic complete subgraph on $k$ vertices. This fact can be applie
 d to deduce Schur's Theorem\, which says that there exists a least positiv
 e integer $S_r(k)$ such that every $r$-coloring of $\\{1\,2\,\\dots\,N\\}$
  for $N\\geq S_r(k)$ yields a monochromatic solution to the equation $x_1+
 x_2+\\cdots+x_{k-1}=x_k$. Here we discuss new findings related to these tw
 o classical results. First\, we derive explicit upper bounds on $R_r(k)$\,
  established through the pigeonhole principle and careful bookkeeping\, th
 at improve upon previously documented bounds. Second\, we present an exten
 sion of Schur's Theorem to higher-dimensional integer lattices\, with the 
 additional restriction that the vectors on the left hand side of the equat
 ion are linearly independent. \n\nThis includes joint work with six (at th
 e time) Millsaps College undergraduate students: Vishal Balaji\, Powers La
 mb\, Andrew Lott\, Dhruv Patel\, Sakshi Singh\, and Christine Rose Ward.\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Russell Jay Hendel (Towson University)
DTSTART:20220526T200000Z
DTEND:20220526T202500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/43
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/43/">A system of 4 simultaneous recursions: Generalization of  Ledin-S
 hannon-Ollerton</a>\nby Russell Jay Hendel (Towson University) as part of 
 Combinatorial and additive number theory (CANT 2022)\n\n\nAbstract\nThis p
 aper further generalizes a recent result of Shannon and Ollerton who resur
 rected an old identity due to Ledin.  \nThis paper generalizes the Ledin-S
 hannon-Ollerton result to all metallic sequences. The results give closed 
 formulas for the sum of products of powers of the first $n$ integers with 
 the first $n$ members of the metallic sequence. \nThree key innovations of
  this paper are (i) reducing the proof of the generalization to the soluti
 on of  a system of 4 simultaneous recursions\;\n(ii) skillful use of the s
 hift operation to prove equality of polynomials\; and (iii) new OEIS seque
 nces\narising from the coefficients of  the four polynomial\nfamilies sati
 sfying the four simultaneous recursions.\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ariane Masuda (New York City College of Technology\, CUNY)
DTSTART:20220526T203000Z
DTEND:20220526T205500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/44
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/44/">Redei permutations with the same cycle structure</a>\nby Ariane M
 asuda (New York City College of Technology\, CUNY) as part of Combinatoria
 l and additive number theory (CANT 2022)\n\n\nAbstract\nPermutation polyno
 mials over finite fields have been extensively studied over the past decad
 es. Among the major challenges in this area are the questions concerning t
 heir cycle structures as they capture relevant properties\, both theoretic
 ally and practically. In this talk we focus on a family of permutation pol
 ynomials\, the so called R\\'edei permutations. Although their cycle struc
 tures are known\, there are other related questions that can be investigat
 ed. For example\, when do two R\\'edei permutations have the same cycle st
 ructure? We give a characterization of such pairs\, and present explicit f
 amilies of R\\'edei permutations with the same cycle structure. We also di
 scuss some results regarding R\\'edei permutations with a particularly sim
 ple cycle structure\, consisting of $1$- and $j$-cycles only\, when $j$ is
  $4$ or a prime number. The case $j = 2$ is specially important in some ap
 plications. We completely describe R\\'edei involutions with a prescribed 
 cycle structure\, and show that the only R\\'edei permutations with a uniq
 ue cycle structure are the involutions. \n\nThis is joint work with Julian
 e Capaverde and Virg\\'inia Rodrigues.\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sean Prendiville (Lancaster University\, UK)
DTSTART:20220527T130000Z
DTEND:20220527T132500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/45
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/45/">Adapting the circle method for colourings</a>\nby Sean Prendivill
 e (Lancaster University\, UK) as part of Combinatorial and additive number
  theory (CANT 2022)\n\n\nAbstract\nFix your favourite Diophantine equation
 . If each integer is coloured red\, blue or green\, how many solutions to 
 your equation have all variables the same colour? We discuss how to adapt 
 the Hardy-Littlewood circle method to yield a lower bound in certain probl
 ems of this flavour.\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ajmain Yamin (CUNY  Graduate Center)
DTSTART:20220527T133000Z
DTEND:20220527T135500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/46
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/46/">The exceptional automorphism of $S_6$ explained with colored maps
 </a>\nby Ajmain Yamin (CUNY  Graduate Center) as part of Combinatorial and
  additive number theory (CANT 2022)\n\n\nAbstract\nAmong all symmetric gro
 ups\, $S_6$ is the only one with a nontrivial outer automorphism\, \nIn th
 is talk\, I will describe a new way to understand the exotic embedding of 
 $S_5 \\hookrightarrow S_6$ in terms of $5$-colored complete regular maps o
 n the torus.  This provides a visual explanation for the existence of the 
 exceptional automorphism of $S_6$.\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/46/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Paolo Leonetti (Universita ``Luigi Bocconi''\, Milano\, Italy)
DTSTART:20220527T140000Z
DTEND:20220527T142500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/47
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/47/">The G.C.D. of $n$ and the $n$th Fibonacci number</a>\nby Paolo Le
 onetti (Universita ``Luigi Bocconi''\, Milano\, Italy) as part of Combinat
 orial and additive number theory (CANT 2022)\n\n\nAbstract\nLet $(F_n)_{n 
 \\geq 1}$ be the sequence of Fibonacci numbers\, defined as usual by $F_1 
 = F_2 = 1$ and $F_{n + 2} = F_{n + 1} + F_n$ for all positive integers $n$
 \; and let $\\mathcal{A}$ be the set of all integers of the form $\\gcd(n\
 , F_n)$\, for some positive integer $n$.\nIn this talk we shall illustrate
  the following result on $\\mathcal{A}$.\n\n\\noindent\n\\textbf{Theorem.}
  \\textit{For all $x \\geq 2$\, we have\n\\begin{equation*}\n\\#\\mathcal{
 A}(x) \\gg \\frac{x}{\\log x} .\n\\end{equation*}\nOn the other hand\, $\\
 mathcal{A}$ has zero asymptotic density.}\nThe proofs rely on a result of 
 Cubre and Rouse (PAMS\, 2014) which gives\, for each positive integer $n$\
 , an explicit formula for the density of primes $p$ such that $n$ divides 
 the rank of appearance of $p$\, that is\, the smallest positive integer $k
 $ such that $p$ divides $F_k$.\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/47/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bartosz Sobolewski (Jagiellonian University\, Krakow\, Poland)
DTSTART:20220527T150000Z
DTEND:20220527T152500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/48
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/48/">Monochromatic arithmetic progressions in binary words associated 
 with pattern sequences</a>\nby Bartosz Sobolewski (Jagiellonian University
 \, Krakow\, Poland) as part of Combinatorial and additive number theory (C
 ANT 2022)\n\n\nAbstract\nLet $e_v(n)$ denote the number of occurrences of 
 a pattern $v$ in the binary expansion of $n \\in \\mathbb{N}$. In the talk
  we consider monochromatic arithmetic progressions in the class of words $
 (e_v(n) \\bmod{2})_{n \\geq 0}$ over $\\{0\,1\\}$\, which includes the Thu
 e--Morse word $\\mathbf{t}$ ($v=1$) and a variant of the Rudin--Shapiro wo
 rd $\\mathbf{r}$ ($v=11$). So far\, the problem of exhibiting long progres
 sions and finding an upper bound on their length has mostly been studied f
 or $\\mathbf{t}$ and certain generalizations. We show that analogous resul
 ts hold for $\\mathbf{r}$. In particular\, we prove that a monochromatic a
 rithmetic progression of difference $d \\geq 3$ starting at $0$ in $\\math
 bf{r}$ has length at most $(d+3)/2$\, with equality infinitely often. We a
 lso compute the maximal length of progressions of differences $2^k-1$ and 
 $2^k+1$.\nSome weaker results for a general pattern $v$ are provided as we
 ll.\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/48/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maciej Ulas (Jagiellonian University\, Krakow\, Poland)
DTSTART:20220527T153000Z
DTEND:20220527T155500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/49
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/49/">Solutions of certain meta-Fibonacci recurrences</a>\nby Maciej Ul
 as (Jagiellonian University\, Krakow\, Poland) as part of Combinatorial an
 d additive number theory (CANT 2022)\n\n\nAbstract\nWe investigate the sol
 utions of certain meta-Fibonacci recurrences of the form $f(n)=f(n-f(n-1))
 +f(n-2)$ for various sets of initial conditions. In the case when $f(n)=1$
  for $n\\leq 1$\, we prove that the resulting integer sequence is closely 
 related to the function counting binary partitions of a certain type (inde
 pendently of the value of $f(2)\\in\\mathbb{N}$). \n\nThe talk is based on
  a joint work with Bartosz Sobolewski.\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/49/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daodao Yang (Graz University of Technology\, Austria)
DTSTART:20220527T170000Z
DTEND:20220527T172500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/50
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/50/">Extreme values of derivatives of the Riemann zeta function\, log-
 type GCD sums\, and spectral norms</a>\nby Daodao Yang (Graz University of
  Technology\, Austria) as part of Combinatorial and additive number theory
  (CANT 2022)\n\n\nAbstract\nFirst I will recall the research on greatest c
 ommon divisor (GCD) sums and   extreme values of  the Riemann zeta functio
 n. The motivation for the study and the connection between the two problem
 s will be discussed.\n Then I will explain how to establish lower bounds f
 or maximums of $|\\zeta^{(\\ell)}\\left(\\sigma+it\\right)|$ when $\\sigma
  \\in [\\frac{1}{2}\, 1]$\, $\\ell \\in \n$.  One of my results states tha
 t   as $T \\to \\infty$\,  uniformly for all positive integers $\\ell \\le
 qslant (\\log_3 T) / (\\log_4 T)$\, we have\n$  \n\\max_{T\\leqslant t\\le
 qslant 2T}\\left|\\zeta^{(\\ell)}\\left(1+it\\right)\\right| \\geqslant \\
 left(\\mathbf Y_{\\ell}+ o\\left(1\\right)\\right)\\left(\\log_2 T \\right
 )^{\\ell+1} $\, where $\\mathbf Y_{\\ell} = \\int_0^{\\infty} u^{\\ell} \\
 rho (u) du$\, and $\\rho(u)$ denotes the Dickman function. This  generaliz
 es results of Bohr-Landau and Littlewood on $\\left|\\zeta\\left(1+it\\rig
 ht)\\right|$ in 1910s.  The tools are Soundararajan's resonance methods an
 d ingredients are certain combinatorial optimization problems. On the othe
 r hand\, assuming the Riemann hypothesis\, we have $|\\zeta^{(\\ell)}\\lef
 t(1+it\\right)| \\ll_{\\ell}\\left(\\log \\log t\\right)^{\\ell+1}$.\nThen
  I will talk on the log-type GCD sums $\\Gamma^{(\\ell)}_{\\sigma}(N)$\, w
 hich I define it as $\\Gamma_{\\sigma}^{(\\ell)}(N):\\\,= \\sup_{|\\mathca
 l{M}| = N} \\frac{1}{N}\\sum_{m\, n\\in \\mathcal{M}} \\frac{(m\,n)^{\\sig
 ma}}{[m\,n]^{\\sigma}}\\log^{\\ell} \\left(\\frac{m}{(m\,n)}\\right)\\log^
 {\\ell}\\left(\\frac{n}{(m\,n)}\\right)\,$\nwhere the supremum is taken ov
 er all subsets   $\\mathcal{M} \\subset \\mathbb N$ with size $N$.\nI will
  explain how $\\Gamma^{(\\ell)}_{\\sigma}(N)$ can be related to $|\\zeta^{
 (\\ell)}(1+it)|$ and how to prove that $\\left(\\log\\log N\\right)^{2+2\\
 ell} \\ll _{\\ell}\\Gamma^{(\\ell)}_1(N)\\ll_{\\ell} \\left(\\log \\log N\
 \right)^{2+2\\ell}$\, which generalizes Gál's theorem (corresponding to t
 he case $\\ell = 0$). The lower bounds could be used to produce large valu
 es of $|\\zeta^{(\\ell)}\\left(1+it\\right)|$.   Using a random model for 
 the zeta function via methods  of Lewko-Radziwiłł\,   upper bounds for s
 pectral norms on $\\alpha$-line are established\,  when $\\alpha \\to 1^{-
 }$ with certain fast rates. As a corollary\,  upper bounds of correct orde
 r of   the log-type GCD sums are established.\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/50/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Filip Gawron (Jagiellonian University\, Poland)
DTSTART:20220527T173000Z
DTEND:20220527T175500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/51
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/51/">Sign behavior of sums of weighted numbers of partitions</a>\nby F
 ilip Gawron (Jagiellonian University\, Poland) as part of Combinatorial an
 d additive number theory (CANT 2022)\n\n\nAbstract\nLet $A$ be a subset of
  the positive integers. By an $A$-partition of $n$ we\nunderstand the repr
 esentation of $n$ as a sum of elements from the set $A$. For\ngiven $i$\, 
 $n\\in \\mathbb{N}$\, by $c_A(i\,n)$ we denote the number of $A$-partition
 s of $n$ with\nexactly $i$ parts. In the talk I will describe several resu
 lts concerning the sign behaviour\nof the sequence $S_{A\,k}(n) = \\sum_{i
 =0}^n(-1)^i i^k c_A(i\, n)$\, for fixed $k\\in \\mathbb{N}$.   I will focu
 s on the periodicity of the sequence of signs for different forms of $A$. 
 Finally\, I will also mention some conjectures and questions that arose na
 turally during our research.\n\nThe talk is based on a joint work with Mac
 iej Ulas (Jagiellonian University).\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/51/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anurag Sahay (University of Rochester)
DTSTART:20220527T180000Z
DTEND:20220527T182500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/52
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/52/">Moments of the Hurwitz zeta function with rational shifts</a>\nby
  Anurag Sahay (University of Rochester) as part of Combinatorial and addit
 ive number theory (CANT 2022)\n\n\nAbstract\nThe Hurwitz zeta function is 
 a shifted integer analogue of the Riemann zeta function\, for shift parame
 ters $0 < \\alpha \\leqslant 1$. We consider the moments of the Hurwitz ze
 ta function on the critical line $\\Re{s} = 1/2$ for rational shifts $\\al
 pha = a/q$. In this case\, the Hurwitz zeta function decomposes as a linea
 r combination of Dirichlet $L$-functions\, which leads us into investigati
 ng moments of products of $L$-functions.\n\nIf time permits\, we will brie
 fly discuss these moments for irrational shift parameters $\\alpha$\, whic
 h shall dovetail into Trevor Wooley's talk on our joint work with Winston 
 Heap.\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/52/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ognian Trifonov (University of South Carolina)
DTSTART:20220527T183000Z
DTEND:20220527T185500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/53
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/53/">Lattice points close to ovals\, arcs\, and helixes</a>\nby Ognian
  Trifonov (University of South Carolina) as part of Combinatorial and addi
 tive number theory (CANT 2022)\n\n\nAbstract\nIn 1972 Schinzel showed that
  the largest distance between three lattice points on a circle of radius $
 R$ \nis at least $\\sqrt[3]{2} R^{1/3}$. We generalize Schinzel's result t
 o ovals and arcs with bounded curvature in the plane  and lattice points c
 lose to the curve.\nFurthermore\, we extend the result to the case of affi
 ne lattices. Finally\, we obtain similar results when the curve is a helix
  in three dimensional space.\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/53/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Brad Isaacson (New York City College of Technology (CUNY))
DTSTART:20220527T190000Z
DTEND:20220527T192500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/54
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/54/">On a polynomial reciprocity theorem of Carlitz</a>\nby Brad Isaac
 son (New York City College of Technology (CUNY)) as part of Combinatorial 
 and additive number theory (CANT 2022)\n\n\nAbstract\nCarlitz proved a pow
 erful reciprocity theorem for generalized Dedekind-Rademacher sums.  Among
  its many consequences was an interesting polynomial reciprocity theorem w
 hich holds under a certain restriction of its parameters.  Carlitz remarke
 d that it was unclear how this restriction could be removed.  In this talk
 \, we remove this restriction and obtain a generalization of Carlitz's pol
 ynomial reciprocity theorem.\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/54/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Faye Jackson (University of Michigan)
DTSTART:20220527T200000Z
DTEND:20220527T202500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/56
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/56/">The Generalized Bergman game</a>\nby Faye Jackson (University of 
 Michigan) as part of Combinatorial and additive number theory (CANT 2022)\
 n\n\nAbstract\nP. Baird-Smith A. Epstein\, K. Flint\, and S. J. Miler\n(20
 18) created the \\emph{Zeckendorf Game}\, a two-player game which takes\na
 s an input a positive integer $n$ and\, using moves related to the\nFibona
 cci recurrence relation\, outputs the unique decomposition of $n$\ninto a 
 sum of non-consecutive Fibonacci numbers. Following this work and\nthat of
  G. Bergman (1957)\, which proved the existence and uniqueness of\nsuch $\
 \varphi$-decompositions\, we formulate the \\emph{Bergman Game} which\nout
 puts the unique decomposition of $n$ into a sum of non-consecutive\npowers
  of $\\varphi$\, the golden mean.\n\nWe then formulate \\emph{Generalized 
 Bergman Games}\, which use moves based\non an arbitrary non-increasing pos
 itive linear recurrence relation and\noutput the unique decomposition of $
 n$ into a sum of non-adjacent powers\nof $\\beta$\, where $\\beta$ is the 
 dominating root of the characteristic\npolynomial of the chosen recurrence
  relation. We prove that the longest\npossible Generalized Bergman game on
  an initial state $S$ with $n$\nsummands terminates in $\\Theta(n^2)$ time
 \, and we also prove that the\nshortest possible Generalized Bergman game 
 on an initial state terminates\nbetween $\\Omega(n)$ and $O(n^2)$ time. We
  also show a linear bound on the\nmaximum length of the tuple used through
 out the game.\n\nThis is joint work with Benjamin Baily\, Justine Dell\, I
 rfan Durmic\, Henry\nFleischmann\, Isaac Mijares\, Steven J. Miller\, Etha
 n Pesikoff\, Alicia Smith\nReina\, and Yingzi Yang.\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/56/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mikhail Gabdullin (Steklov Mathematical Institute\, Moscow\, Russi
 a)
DTSTART:20220527T143000Z
DTEND:20220527T145500Z
DTSTAMP:20260424T222003Z
UID:CANT2022/57
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/CANT2
 022/57/">A conjecture of Cilleruelo and Cordoba and divisors in a short in
 terval</a>\nby Mikhail Gabdullin (Steklov Mathematical Institute\, Moscow\
 , Russia) as part of Combinatorial and additive number theory (CANT 2022)\
 n\n\nAbstract\nLet $E(A)=\\#\\{(a_1\,a_2\,a_3\,a_4)\\in A^4: a_1+a_2=a_3+a
 _4\\}$ denote the additive energy of a set $A\\subset \n$\, and let $\\mat
 hbb{T}=\\R/\\Z$ and $\\|f\\|_4=\\left(\\int_{\\mathbb{T}}|f(t)|^4dt\\right
 )^{1/4}$. It is well-known that \n$$\nE(\\{n^2: n\\leq N\\})=\\left\\|\\su
 m_{n\\leq N}e^{2\\pi in^2x}\\right\\|_4^4 \\asymp N^2\\log N\,\n$$\nwhile 
 we trivially have $E(A)\\geq |A|^2$. In 1992\, J. Cilleruelo and A. Cordob
 a proved that $E(\\{n^2: N\\leq n\\leq N+N^{\\gamma}\\})\\asymp N^{2\\gamm
 a}$ for any $\\gamma\\in (0\,1)$\, and conjectured a much more general bou
 nd (again\, for any $\\gamma\\in(0\,1)$)\n$$\n\\left\\|\\sum_{N\\leq n\\le
 q N+N^{\\gamma}}a_ne^{2\\pi in^2x}\\right\\|_4\\leq C(\\gamma)\\left(\\sum
 _{N\\leq n\\leq N+N^{\\gamma}}|a_n|^2\\right)^{1/2}.\n$$\nWhile this bound
  is easy to prove for $\\gamma\\leq 1/2$\, it seems to be open for any $\\
 gamma>1/2$. We prove this for all $\\gamma<\\frac{\\sqrt5-1}{2}=0.618...$ 
 and present a connection between this problem and a conjecture of I. Ruzsa
 : for any $\\epsilon>0$ there exists $C(\\epsilon)>0$ such that any positi
 ve integer $N$ has at most $C(\\epsilon)$ divisors in the interval $[N^{1/
 2}\, N^{1/2}+N^{1/2-\\epsilon}]$.\n
LOCATION:https://stable.researchseminars.org/talk/CANT2022/57/
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