On sums of subsets of a set of integers |
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Authors: | N. Alon G. Freiman |
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Affiliation: | (1) School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv, Israel |
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Abstract: | Forr2 letp(n, r) denote the maximum cardinality of a subsetA ofN={1, 2,...,n} such that there are noBA and an integery withb=yr. It is shown that for any>0 andn>n(), (1+o(1))21/(r+1)n(r–1)/(r+1)p(n, r)n+2/3 for allr5, and that for every fixedr6,p(n, r)=(1+o(1))·21/(r+1)n(r–1)/(r+1) asn. Letf(n, m) denote the maximum cardinality of a subsetA ofN such that there is noBA the sum of whose elements ism. It is proved that for 3n6/3+mn2/20 log2n andn>n(), f(n, m)=[n/s]+s–2, wheres is the smallest integer that does not dividem. A special case of this result establishes a conjecture of Erds and Graham.Research supported in part by Allon Fellowship, by a Bat-Sheva de Rothschild Grant and by the Fund for Basic Research administered by the Israel Academy of Sciences. |
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Keywords: | 10 A 50 10 B 35 10 J 10 |
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