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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 758.11007

**Autor: ** Erdös, Paul; Sárközy, A.

**Title: ** Arithmetic progressions in subset sums. (In English)

**Source: ** Discrete Math. 102, No.3, 249-264 (1992).

**Review: ** For any set *A* of positive integers, denote by *P*(*A*) the set of positive integers n which can be expressed as a sum of distinct elements of *A*. Let u = F(N,t) be the greatest integer u such that for every *A*\subset**{**1,2,...,N**}** with |*A*| = t, the set *P*(*A*) contains u consecutive multiples of a positive integer d: **{**(x+1)d,(x+2)d,...,(x+u)d**}**\subset *P*(*A*), for some x and d, and let v = G(N,t) be the greatest integer v such that for every *A*\subset**{**1,2,...,N**}** with |*A*| = t, the set *P*(*A*) contains an arithmetic progression of length v. It is clear that F(N,t) \leq G(N,t) for all N,t.

Extending earlier work of Sárközy, the authors prove:

I. If N \geq N_{0} and 18(log N)^{2} < t \leq N. Then F(N,t) > ^{1}/_{18} {t\over {(log N)^{2}}}.

II. (i). If N > N_{0} and c log N < t < ^{1}/_{3} N^{1/3}, then F(N,t) < 16 {t\over{log N}} log ({t\over{log N}}).\par\strut \phantom{II.} (ii). If \epsilon > 0 and t_{0}(\epsilon) < t < (1- \epsilon)N^{ ½}, then F(N,t) < (1+\epsilon)t.

III. (i). If N > N_{0} and \exp(2(log N)^{ ½}) < t < N^{ ½}, then G(N,t) < t\exp**(**4**max****(**{{log N}\over {log t}},{{(log t)^{2}} \over {log N}}**)****)**. \phantom{III.} (ii). For all t_{0} < t < ^{1}/_{2} N^{ ½}, G(N,t) < 2t^{3/2}.

Finally, upper and lower bounds are also obtained for certain other functions associated with 3 term arithmetic progressions.

**Reviewer: ** M.Nair (Glasgow)

**Classif.: ** * 11B25 Arithmetic progressions

**Keywords: ** subset sums; sums of distinct elements of a sequence; arithmetic progression

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