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SUMMARY:Ethan Patrick White (University of British Columbia)
DTSTART:20220524T203000Z
DTEND:20220524T205500Z
DTSTAMP:20260423T011340Z
UID:CANT2022/13
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2022/13/
 ">Erdos' minimum overlap problem</a>\nby Ethan Patrick White (University o
 f British Columbia) as part of Combinatorial and additive number theory (C
 ANT 2022)\n\n\nAbstract\nIn 1955 Erd\\H{o}s posed the following problem. L
 et $n$ be a positive integer and $A\,B \\subset [2n    A weighted generali
 zation of classical zero-sum constants\nwas introduced by Adhikari {\\it e
 t 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\, we take 
 up the problem of determining the exact values and providing bounds of the
  weighted Davenport constant of $\\mathbb {Z}_n$  \nwith some new weight s
 ets.\n\nNext\, we consider a weighted generalization of the {\\it the 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 defined 
 to be the  least positive integer $t$ such that\nany sequence $S$ of $t$ e
 lements 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 integers $
 k\\geq 2$ and $A=\\{1\,2\\}$\,\nwhere $G$ 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 the problem of charac
 terizing all the weighted {\\it zero-sum free sequences} over $\\mathbb {Z
 }_n$ of specific lengths for the particular weight sets under consideratio
 n.\n\nThis work was joint with Sukumar Das Adhikari and partly with Md Ibr
 ahim 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 b
 e the number of solutions $(a\,b) \\in A  \\times B$ to $a-b = k$. Estimat
 e the size of the function\n\\[ M(n) = \\min_{A\\cup B = [2n]} \\max_{-2n<
 k<2n} M_k\,\\]\nwhere the minimum is taken over all partitions of $[2n]$ i
 nto equal-sized sets. Many upper and lower estimates were obtained over th
 e 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://researchseminars.org/talk/CANT2022/13/
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