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The inverse Erd˝ os-Heilbronn problem

unpublished

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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