On sums of subsets of a set of integers |
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Authors: | N Alon G Freiman |
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Institution: | (1) School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv, Israel |
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Abstract: | Forr 2 letp(n, r) denote the maximum cardinality of a subsetA ofN={1, 2,...,n} such that there are noB A and an integery with
b=y
r. 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 allr 5, and that for every fixedr 6,p(n, r)=(1+o(1))·21/(r+1)
n
(r–1)/(r+1) asn![rarr](/content/f87724506p641820/xxlarge8594.gif) . Letf(n, m) denote the maximum cardinality of a subsetA ofN such that there is noB A the sum of whose elements ism. It is proved that for 3n
6/3+![epsiv](/content/f87724506p641820/xxlarge603.gif) m n
2/20 log2
n 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 Erd s 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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