An Erdős problem on random subset sums in finite abelian groups

TL;DR

This work resolves Erdős’s conjecture in the prime-order setting by proving a quantitative lower bound $f(p) \ge \log_2 p + (\tfrac{1}{2\log 2}+o(1))\log\log p$ for large primes $p$, showing that the universal bound $f(N) \le \log_2 N + o(\log\log N)$ cannot hold. The authors reduce the subset-sum problem to an iid model, and then employ a Poisson-like moment analysis together with a second-moment method to demonstrate most primes $p$ violate the conjectured threshold. A key technical component is establishing that the count of missed subset-sums behaves like a Poisson variable with mean $\lambda = \frac{M}{p}$, where $M=2^k-1$ and $k$ is chosen near $\log_2 p$. The work combines probabilistic methods, linear-algebra over finite fields, and delicate combinatorial estimates to derive the lower bound, with the result highlighting that the asymptotic second-order term in $f(p)$ can be strictly larger than previously conjectured for primes. The paper also documents an AI-assisted research workflow, illustrating an example of human–AI collaboration in theoretical combinatorics and probabilistic method applications.

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, Erdős and Rényi proved that for all $N$, $$ f(N) \le \log_2 N + \left(\frac{1}{\log 2}+o(1)\right)\log\log N. $$ Erdős 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 $p$, $$ f(p)\ge \log_2 p+\left(\frac{1}{2\log 2}+o(1)\right)\log\log p. $$ This work is an outcome of human--AI collaboration: the original qualitative proof was generated autonomously by ChatGPT-5.2 Pro, while the quantitative refinement was developed by the authors.

Theorems & Definitions (20)

• Theorem 1.2
• Corollary 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof