{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,6,26]],"date-time":"2022-06-26T18:06:04Z","timestamp":1656266764194},"reference-count":20,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2012,4,23]],"date-time":"2012-04-23T00:00:00Z","timestamp":1335139200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2012,7]]},"abstract":"<jats:p>Given a finite subset <jats:italic>A<\/jats:italic> of an abelian group <jats:italic>G<\/jats:italic>, we study the set <jats:italic>k<\/jats:italic> \u2227 <jats:italic>A<\/jats:italic> of all sums of <jats:italic>k<\/jats:italic> distinct elements of <jats:italic>A<\/jats:italic>. In this paper, we prove that |<jats:italic>k<\/jats:italic> \u2227 <jats:italic>A<\/jats:italic>| \u2265 |<jats:italic>A<\/jats:italic>| for all <jats:italic>k<\/jats:italic> \u2208 {2,.\u00a0.\u00a0.,|<jats:italic>A<\/jats:italic>| \u2212 2}, unless <jats:italic>k<\/jats:italic> \u2208 {2, |<jats:italic>A<\/jats:italic>| \u2212 2} and <jats:italic>A<\/jats:italic> is a coset of an elementary 2-subgroup of <jats:italic>G<\/jats:italic>. Furthermore, we characterize those finite sets <jats:italic>A<\/jats:italic> \u2286 <jats:italic>G<\/jats:italic> for which |<jats:italic>k<\/jats:italic> \u2227 <jats:italic>A<\/jats:italic>| = |<jats:italic>A<\/jats:italic>| for some <jats:italic>k<\/jats:italic> \u2208 {2,.\u00a0.\u00a0.,|<jats:italic>A<\/jats:italic>| \u2212 2}. This result answers a question of Diderrich. Our proof relies on an elementary property of proper edge-colourings of the complete graph.<\/jats:p>","DOI":"10.1017\/s0963548312000168","type":"journal-article","created":{"date-parts":[[2012,4,23]],"date-time":"2012-04-23T15:00:42Z","timestamp":1335193242000},"page":"582-596","source":"Crossref","is-referenced-by-count":4,"title":["<i>k<\/i>-Sums in Abelian Groups"],"prefix":"10.1017","volume":"21","author":[{"given":"BENJAMIN","family":"GIRARD","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"SIMON","family":"GRIFFITHS","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"YAHYA OULD","family":"HAMIDOUNE","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2012,4,23]]},"reference":[{"key":"S0963548312000168_ref1","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548398003411"},{"key":"S0963548312000168_ref13","doi-asserted-by":"publisher","DOI":"10.1017\/S096354830000451X"},{"key":"S0963548312000168_ref7","doi-asserted-by":"publisher","DOI":"10.1007\/BF02761530"},{"key":"S0963548312000168_ref5","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s1-10.37.30"},{"key":"S0963548312000168_ref19","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511755149"},{"key":"S0963548312000168_ref4","first-page":"99","article-title":"Recherches sur les nombres","volume":"9","author":"Cauchy","year":"1813","journal-title":"J. \u00c9cole Polytechnique"},{"key":"S0963548312000168_ref18","first-page":"87","volume-title":"Combinatorial Number Theory and Additive Group Theory","author":"Ruzsa","year":"2009"},{"key":"S0963548312000168_ref8","first-page":"41","article-title":"Theorem in the additive number theory","volume":"10","author":"Erd\u0151s","year":"1961","journal-title":"Bull. 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