Recent advances in generalized MSTD problems and Zeckendorf games

Abstract: We report on two areas studied this summer in Miller's REU: Generalized MSTD Problems and Zeckendorf Games.

1. A finite integer subset $A \subseteq \mathbb{Z}$ is classified as a More Sums Than Differences (MSTD, or sum-dominant) set when it produces strictly more pairwise sums than differences, satisfying $|A+A| > |A-A|$. Motivated by the structural density of these integer sets, we generalize this phenomenon to subsets $A$ of a finite group $G$ by comparing the cardinality of the product set $AA$ against the quotient set $AA^{-1}$. To evaluate global group behavior, we analyze the weighted difference across all possible subsets, defined as $$W(G) = \sum_{A \subseteq G} (|AA| - |AA^{-1}|).$$ Using a combination of combinatorial techniques, graph theory, and representation theory, we prove that $W(G)$ is strictly negative for all finite abelian groups—establishing them as inherently quotient-dominant—and we successfully extend these structural findings to characterize select non-abelian groups.

2. Zeckendorf proved every integer can be written uniquely as a sum of non-adjacent Fibonacci numbers $\{F_n\}$. Using the Fibonacci recurrence, Miller created the Zeckendorf game. Starting with $n$ copies of $F_1$, a player either replaces a copy of $F_i$ and $F_{i-1}$ with $F_{i+1}$, or splits two copies of $F_i$ into $F_{i+1}$ and $F_{i-1}$ (with $F_2$ splitting to $F_3$ and $F_1$). All games terminate in the Zeckendorf decomposition of $n$; whomever moves last wins. A non-constructive proof exists that Player Two has a winning strategy for all $n > 2$. We discuss current work on a variety of generalizations, including binary decompositions, first to reach the largest summand wins, and higher dimensional analogues.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory seminar (CANT 2026)