Subset Sums
نویسنده
چکیده
IAl > ((l/k) + E)n then there is a subset B L A such that 0 < 1 BI 0, let snd(nt) denote the smallest integer that does not divide PI. We prove that for every I-: > 0 there is a constant c = ~(8:) z I, such that for every n > 0 and every rn, n ' +' 6 WI < n'llog'n every set A E j I, Z,..., II) of cardinality IAl > c.n/snd(m) contains a subset Bcl-.4 so that ChcB h =m. This is best possible, up to the constant C. In particular it implies that for every II there is an m such that every set A c (l,..., II jof cardinality IAl > cx/log II contains a subset BG A so that xhtS h = ,n, thus settling a problem of Erdds and Graham. Let n be a positive integer and put N = i 1, 2,..., II) For IPZ 3 1 let J'(K WZ) denote the maximum cardinality of a set A i N that contains no subset BEA so that ChtB h = m. Here we first show that f(n. 2n)=($+O(l))~Iz (1.1) (as n + co). This settles a problem of Erdos and Graham [El. To establish (1.1) we prove the following result about subset sums in the abelian group z,. THEOREM 1.1. For every-fixed c > 0 and k > 1, lf II > n,(k, F) and A c Z,, satisfies IAl > ((l/k) + e) n then there is a subset BE A such that O<(BI <k andC,..h=O. The case k = 3 of this theorem solves a problem of Stalley [S]. Clearly
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Inverse Theorems for Subset Sums
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