The famous Erd˝ os-Heilbronn conjecture (first proved by Dias da Silva and Hami-doune in 1994) asserts that if A is a subset of Z/pZ, the cyclic group of the integers modulo a prime p, then |A + A| 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 ⊂ Z/pZ contains at least 5 elements and |A + A| 2 |A| − 3 < p, then A must be an arithmetic progression. We consider a large prime p and

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... igate the following more general question: what is the structure of sets A ⊂ Z/pZ such that |A + A| (2 + ǫ) |A|? Our main result is an asymptotically complete answer to this question: there exists a function δ(p) = o(1) such that if 200 < |A| (1 − ǫ ′)p/2 and if |A + A| (2 + ǫ) |A|, where ǫ ′ − ǫ δ > 0, then A is contained in an arithmetic progression of length |A + A| − |A| + 3. With the extra assumption that |A| (1 2 − 1 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 Erd˝ os-Heilbronn conjecture and an inverse Erd˝ os-Heilbronn theorem.