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**Zentralblatt MATH**

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

**Zbl.No: ** 633.10047

**Autor: ** Erdös, Paul; Nathanson, Melvyn B.; Sárközy, András

**Title: ** Sumsets containing infinite arithmetic progressions. (In English)

**Source: ** J. Number Theory 28, No.2, 159-166 (1988).

**Review: ** The authors prove some quantitative results on infinite arithmetic progressions contained in sumsets of sets A (of nonnegative integers) of positive lower asymptotic density w. If k is the smallest integer such that k \geq 1/w, it is proved (i) that there is an infinite progression with difference at most k+1 such that every term of the progression can be written as a sum of exactly k^{2}-k distinct terms of A, (ii) there is an infinite arithmetic progression with difference at most k^{2}-k such that every term of the progression can be written as a sum of exactly k+1 distinct terms of A. A solution is also shown to the infinite analog of two problems of Erdös and R. Freud on the representation of powers of 2 and square-free numbers as bounded sums of distinct elements chosen from a set with specified positive density.

**Reviewer: ** St.Porubský

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

11B83 Special sequences of integers and polynomials

**Keywords: ** infinite arithmetic progressions; sumsets; representation of powers of 2 and square-free numbers

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