An Erd\H{o}s problem on random subset sums in finite abelian groups

TL;DR

This paper investigates the minimal size of random subsets in finite abelian groups needed to likely generate the entire group via subset sums, confirming Erdős's conjecture on the lower bound for prime groups.

Contribution

It proves a new lower bound for the subset size in prime groups, confirming Erdős's conjecture on the optimality of the known upper bound.

Findings

Established a lower bound for prime groups matching the conjectured form.

Confirmed Erdős's conjecture on the minimal subset size for generating the entire group.

Extended understanding of subset sum properties in finite abelian groups.

Abstract

Let $f(N)$ denote the least integer $k$ such that, if $G$ is an abelian group of order $N$ and $A \subseteq G$ is a uniformly random $k$-element subset, then with probability at least $\tfrac12$ the subset-sum set $\{ \sum_{x \in S} x : S \subseteq A \}$ equals $G$. In 1965, Erd\H{o}s and R\'{e}nyi proved that for all $N$, $$ f(N) \le \log_2 N + \left(\frac{1}{\log 2}+o(1)\right)\log\log N. $$ Erd\H{o}s later conjectured that this bound cannot be improved to $f(N)\le \log_2 N+o(\log\log N)$. In this paper we confirm this conjecture by showing that, for primes $p$, $$ f(p)\ge \log_2 p+\left(\frac{1}{2\log 2}+o(1)\right)\log\log p. $$