An Erd\H{o}s problem on random subset sums 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 $p$, $$ f(p)\ge \log_2 p+\left(\frac{1}{2\log 2}+o(1)\right)\log\log p. $$