Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Equidistribution of subset sums

Péter Pál Pach

TL;DR

The paper resolves questions of Katona and Makar-Limanov on the equidistribution of $h$-element subset sums in abelian groups by proving asymptotic equidistribution for all abelian groups when the group order $n$ satisfies $n\ge 2h$, and by establishing a growing-$h$ version with an explicit error term. The core method uses an inclusion-exclusion decomposition via partitions of the index set to count distinct coordinates, combined with a gcd-based analysis to isolate uniformly distributed contributions. The main result gives a precise main term $(1/n)\cdot(n!)/(n-h)!$ for the count of representations of each $a\in G$, with a geometric error bound $\le (3/4)^h$ times the main term, and extends to the regime where $h$ grows with $n$, yielding equidistribution for the important case $n=2h$. Collectively, these findings resolve several open problems posed by KML08 and have potential implications for related coding-theoretic questions and combinatorial design.

Abstract

We answer a question of Katona and Makar-Limanov, by showing that in an abelian group of order $2h$ the $h$-element subset sums are asymptotically (as $h\to \infty$) equidistributed. In fact we prove a more general result where the order of the group can be arbitrary, also providing a bound for the ``error term''.

Equidistribution of subset sums

TL;DR

The paper resolves questions of Katona and Makar-Limanov on the equidistribution of -element subset sums in abelian groups by proving asymptotic equidistribution for all abelian groups when the group order satisfies , and by establishing a growing- version with an explicit error term. The core method uses an inclusion-exclusion decomposition via partitions of the index set to count distinct coordinates, combined with a gcd-based analysis to isolate uniformly distributed contributions. The main result gives a precise main term for the count of representations of each , with a geometric error bound times the main term, and extends to the regime where grows with , yielding equidistribution for the important case . Collectively, these findings resolve several open problems posed by KML08 and have potential implications for related coding-theoretic questions and combinatorial design.

Abstract

We answer a question of Katona and Makar-Limanov, by showing that in an abelian group of order the -element subset sums are asymptotically (as ) equidistributed. In fact we prove a more general result where the order of the group can be arbitrary, also providing a bound for the ``error term''.
Paper Structure (5 sections, 5 theorems, 42 equations)

This paper contains 5 sections, 5 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Preliminary lemmas
  3. Proofs
  4. Concluding remarks
  5. Acknowledgements

Key Result

Theorem 1.2

Let $h\geq 3$ be a fixed integer. Let $G$ be an abelian group of order $n\geq 2h$. Then the number of solutions to $x_1+\dots+x_h=a$ with distinct $x_1,\dots,x_h\in G$ is $\frac{1}{n}\cdot \frac{n!}{(n-h)!}+O_h(n^{h/2})$. Therefore, the $h$-element subset sums are asymptotically equidistributed in $

Theorems & Definitions (11)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Lemma 2.1
  • Remark 2.2
  • proof : Proof of Lemma \ref{['lem-incl-excl']}
  • Lemma 2.3
  • proof
  • proof : Proof of Theorem \ref{['thm-small-h']}
  • ...and 1 more