The Inverse Erdős-Heilbronn Problem

Abstract

The famous Erdős-Heilbronn conjecture (first proved by Dias da Silva and Hamidoune in 1994) asserts that if $A$ is a subset of ${\Bbb Z}/p{\Bbb Z}$, the cyclic group of the integers modulo a prime $p$, then $|A\widehat{+}A| \ge \min\{2|A| -3,p\}. $ The bound is sharp, as is shown by choosing $A$ to be an arithmetic progression. A natural inverse result was proven by Karolyi in 2005: if $A\subset {\Bbb Z}/p{\Bbb Z}$ contains at least 5 elements and $|A\widehat{+}A| \le 2|A| -3 < p$, then $A$ must be an arithmetic progression. We consider a large prime $p$ and investigate the following more general question: what is the structure of sets $A \subset {\Bbb Z}/p{\Bbb Z}$ such that $|A\widehat{+}A| \le (2+\epsilon)|A|$? Our main result is an asymptotically complete answer to this question: there exists a function $\delta(p) = o(1)$ such that if $200 < |A| \le (1-\epsilon')p/2$ and if $|A\widehat{+}A| \le (2+\epsilon)|A|$, where $\epsilon' -\epsilon \ge \delta > 0$, then $A$ is contained in an arithmetic progression of length $|A\widehat{+}A| -|A| + 3$. With the extra assumption that $|A| \le ({1\over 2}- {1\over\log^c p} ) p$, our main result has Dias da Silva and Hamidoune's theorem and Karolyi's theorem as corollaries, and thus, our main result provides purely combinatorial proofs for the Erdős-Heilbronn conjecture and an inverse Erdős-Heilbronn theorem.