The asymptotic uniform distribution of subset sums

TL;DR

This paper proves that in large finite abelian groups, the distribution of subset sums becomes uniformly spread out as the group size grows, confirming a long-standing conjecture for a wide range of subset sizes.

Contribution

It establishes the asymptotic uniformity of subset sum distributions in finite abelian groups for a broad range of subset sizes, answering a question posed by Katona and Makar-Limanov.

Findings

Minimum and maximum sizes of subset sum families become asymptotically equal.

Uniform distribution of subset sums as group order tends to infinity.

Results hold for all subset sizes from 4 up to roughly half the group size.

Abstract

Let $G$ be a finite abelian group of order $n$, and for each $a\in G$ and integer $1\le h\le n$ let $\mathcal{F}_a(h)$ denote the family of all $h$-element subsets of $G$ whose sum is $a$. A problem posed by Katona and Makar-Limanov is to determine whether the minimum and maximum sizes of the families $\mathcal{F}_a(h)$ (as $a$ ranges over $G$) become asymptotically equal as $n\rightarrow \infty$ when $h=\left\lfloor\frac{n}{2}\right\rfloor$. We affirmatively answer this question and in fact show that the same asymptotic equality holds for every $4\leq h\leq \left\lfloor\frac{n}{2}\right\rfloor+1$.