{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,3,22]],"date-time":"2024-03-22T00:40:31Z","timestamp":1711068031786},"reference-count":15,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2023,11,1]],"date-time":"2023-11-01T00:00:00Z","timestamp":1698796800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,11,1]],"date-time":"2023-11-01T00:00:00Z","timestamp":1698796800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Combinatorica"],"published-print":{"date-parts":[[2024,4]]},"abstract":"Abstract<\/jats:title>Let A<\/jats:italic> be a subset of the cyclic group $${\\textbf{Z}}\/p{\\textbf{Z}}$$<\/jats:tex-math>\n \n Z<\/mml:mi>\n \/<\/mml:mo>\n p<\/mml:mi>\n Z<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> with p<\/jats:italic> prime. It is a well-studied problem to determine how small |A<\/jats:italic>| can be if there is no unique sum in $$A+A$$<\/jats:tex-math>\n \n A<\/mml:mi>\n +<\/mml:mo>\n A<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, meaning that for every two elements $$a_1,a_2\\in A$$<\/jats:tex-math>\n \n \n a<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \n a<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n \u2208<\/mml:mo>\n A<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, there exist $$a_1',a_2'\\in A$$<\/jats:tex-math>\n \n \n a<\/mml:mi>\n 1<\/mml:mn>\n \u2032<\/mml:mo>\n <\/mml:msubsup>\n ,<\/mml:mo>\n \n a<\/mml:mi>\n 2<\/mml:mn>\n \u2032<\/mml:mo>\n <\/mml:msubsup>\n \u2208<\/mml:mo>\n A<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> such that $$a_1+a_2=a_1'+a_2'$$<\/jats:tex-math>\n \n \n a<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n +<\/mml:mo>\n \n a<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n =<\/mml:mo>\n \n a<\/mml:mi>\n 1<\/mml:mn>\n \u2032<\/mml:mo>\n <\/mml:msubsup>\n +<\/mml:mo>\n \n a<\/mml:mi>\n 2<\/mml:mn>\n \u2032<\/mml:mo>\n <\/mml:msubsup>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and $$\\{a_1,a_2\\}\\ne \\{a_1',a_2'\\}$$<\/jats:tex-math>\n \n \n {<\/mml:mo>\n \n a<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \n a<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n }<\/mml:mo>\n <\/mml:mrow>\n \u2260<\/mml:mo>\n \n {<\/mml:mo>\n \n a<\/mml:mi>\n 1<\/mml:mn>\n \u2032<\/mml:mo>\n <\/mml:msubsup>\n ,<\/mml:mo>\n \n a<\/mml:mi>\n 2<\/mml:mn>\n \u2032<\/mml:mo>\n <\/mml:msubsup>\n }<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Let m<\/jats:italic>(p<\/jats:italic>) be the size of a smallest subset of $${\\textbf{Z}}\/p{\\textbf{Z}}$$<\/jats:tex-math>\n \n Z<\/mml:mi>\n \/<\/mml:mo>\n p<\/mml:mi>\n Z<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> with no unique sum. The previous best known bounds are $$\\log p \\ll m(p)\\ll \\sqrt{p}$$<\/jats:tex-math>\n \n log<\/mml:mo>\n p<\/mml:mi>\n \u226a<\/mml:mo>\n m<\/mml:mi>\n \n (<\/mml:mo>\n p<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \u226a<\/mml:mo>\n \n p<\/mml:mi>\n <\/mml:msqrt>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. In this paper we improve both the upper and lower bounds to $$\\omega (p)\\log p \\leqslant m(p)\\ll (\\log p)^2$$<\/jats:tex-math>\n \n \u03c9<\/mml:mi>\n \n (<\/mml:mo>\n p<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n log<\/mml:mo>\n p<\/mml:mi>\n \u2a7d<\/mml:mo>\n m<\/mml:mi>\n \n (<\/mml:mo>\n p<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \u226a<\/mml:mo>\n \n \n (<\/mml:mo>\n log<\/mml:mo>\n p<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> for some function $$\\omega (p)$$<\/jats:tex-math>\n \n \u03c9<\/mml:mi>\n (<\/mml:mo>\n p<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> which tends to infinity as $$p\\rightarrow \\infty $$<\/jats:tex-math>\n \n p<\/mml:mi>\n \u2192<\/mml:mo>\n \u221e<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. In particular, this shows that for any $$B\\subset {\\textbf{Z}}\/p{\\textbf{Z}}$$<\/jats:tex-math>\n \n B<\/mml:mi>\n \u2282<\/mml:mo>\n Z<\/mml:mi>\n \/<\/mml:mo>\n p<\/mml:mi>\n Z<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> of size $$|B|<\\omega (p)\\log p$$<\/jats:tex-math>\n \n |<\/mml:mo>\n B<\/mml:mi>\n |<\/mml:mo>\n <<\/mml:mo>\n \u03c9<\/mml:mi>\n (<\/mml:mo>\n p<\/mml:mi>\n )<\/mml:mo>\n log<\/mml:mo>\n p<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, its sumset $$B+B$$<\/jats:tex-math>\n \n B<\/mml:mi>\n +<\/mml:mo>\n B<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> contains a unique sum. We also obtain corresponding bounds on the size of the smallest subset of a general Abelian group having no unique sum.<\/jats:p>","DOI":"10.1007\/s00493-023-00069-w","type":"journal-article","created":{"date-parts":[[2023,11,1]],"date-time":"2023-11-01T15:47:42Z","timestamp":1698853662000},"page":"269-298","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On Unique Sums in Abelian Groups"],"prefix":"10.1007","volume":"44","author":[{"given":"Benjamin","family":"Bedert","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,11,1]]},"reference":[{"issue":"3, Special Issu","key":"69_CR1","doi-asserted-by":"publisher","first-page":"343","DOI":"10.1007\/PL00009351","volume":"19","author":"YF Bilu","year":"1998","unstructured":"Bilu, Y.F., Lev, V.F., Ruzsa, I.Z.: Rectification principles in additive number theory. Discrete Comput. 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