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SUMMARY:Steven Miller’s REU: Probability And Number THeory (Williams Col
 lege)
DTSTART:20260713T200000Z
DTEND:20260713T205000Z
DTSTAMP:20260710T111653Z
UID:CANT2026/19
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2026/19/
 ">Recent advances in generalized MSTD problems and Zeckendorf games</a>\nb
 y Steven Miller’s REU: Probability And Number THeory (Williams College) 
 as part of Combinatorial and additive number theory seminar (CANT 2026)\n\
 nLecture held in Science Center in the CUNY Graduate Center (4th floor).\n
 \nAbstract\nWe report on two areas studied this summer in Miller's REU: Ge
 neralized\nMSTD Problems and Zeckendorf Games. \n\n1. A finite integer sub
 set $A \\subseteq\n\\mathbb{Z}$ is classified as a More Sums Than Differen
 ces (MSTD\, or\nsum-dominant) set when it produces strictly more pairwise 
 sums than\ndifferences\, satisfying $|A+A| > |A-A|$. Motivated by the stru
 ctural\ndensity of these integer sets\, we generalize this phenomenon to s
 ubsets\n$A$ of a finite group $G$ by comparing the cardinality of the prod
 uct set\n$AA$ against the quotient set $AA^{-1}$. To evaluate global group
 \nbehavior\, we analyze the weighted difference across all possible subset
 s\,\ndefined as $$W(G) = \\sum_{A \\subseteq G} (|AA| - |AA^{-1}|).$$ Usin
 g a\ncombination of combinatorial techniques\, graph theory\, and represen
 tation\ntheory\, we prove that $W(G)$ is strictly negative for all finite 
 abelian\ngroups—establishing them as inherently quotient-dominant—and 
 we successfully extend these structural findings to characterize select\nn
 on-abelian groups. \n\n2. Zeckendorf proved every integer can be written u
 niquely as a sum of\nnon-adjacent Fibonacci numbers $\\{F_n\\}$. Using the
  Fibonacci recurrence\,\nMiller created the Zeckendorf game. Starting with
  $n$ copies of $F_1$\, a\nplayer either replaces a copy of $F_i$ and $F_{i
 -1}$ with $F_{i+1}$\, or\nsplits two copies of $F_i$ into $F_{i+1}$ and $F
 _{i-1}$ (with $F_2$\nsplitting to $F_3$ and $F_1$). All games terminate in
  the Zeckendorf\ndecomposition of $n$\; whomever moves last wins. A non-co
 nstructive proof\nexists that Player Two has a winning strategy for all $n
  > 2$. We discuss\ncurrent work on a variety of generalizations\, includin
 g binary\ndecompositions\, first to reach the largest summand wins\, and h
 igher\ndimensional analogues.\n
LOCATION:https://researchseminars.org/talk/CANT2026/19/
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