{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,23]],"date-time":"2026-08-23T14:58:43Z","timestamp":1787497123187,"version":"build-2736575974"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>The famous Erd\u0151s-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 $|A\\widehat{+}A| \\le 2|A| -3  &lt; p$, 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 $200 &lt;  |A| \\le (1-\\epsilon')p\/2$ and if $|A\\widehat{+}A| \\le (2+\\epsilon)|A|$, where $\\epsilon' -\\epsilon \\ge \\delta &gt; 0$, 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 Erd\u0151s-Heilbronn conjecture and an inverse Erd\u0151s-Heilbronn theorem.<\/jats:p>","DOI":"10.37236\/189","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T23:27:34Z","timestamp":1578698854000},"source":"Crossref","is-referenced-by-count":4,"title":["The Inverse Erd\u0151s-Heilbronn Problem"],"prefix":"10.37236","volume":"16","author":[{"given":"Van H.","family":"Vu","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Philip Matchett","family":"Wood","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"23455","published-online":{"date-parts":[[2009,8,7]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i1r100\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i1r100\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T21:41:53Z","timestamp":1579297313000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v16i1r100"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,8,7]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2009,1,7]]}},"URL":"https:\/\/doi.org\/10.37236\/189","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,8,7]]},"article-number":"R100"}}