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**Zentralblatt MATH**

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

**Zbl.No: ** 697.10047

**Autor: ** Erdös, Paul; Freiman, Gregory

**Title: ** On two additive problems. (In English)

**Source: ** J. Number Theory 34, No.1, 1-12 (1990).

**Review: ** Let *A* be a set of nonnegative integers. P. Erdös and R. Freud conjectured that if *A* satisfies *A*\subset **{**1,2,...,3n**}** and |*A*| \geq n+1 then there is a power of 2 that can be written as a sum of distinct elements of *A*. Very similarly if *A*\subset **{**1,2,...,4n**}** and |*A*| \geq n+1 then there is a square-free number that can be written as a sum of distinct elements of *A*. Both problems are answered in the affirmative in this paper. The proof is based on the Hardy-Littlewood method and elementary considerations. In this way at least c log n summands from *A* are necessary. Recently, *M. B. Nathanson* and *A. Sárközy* [Acta Arith. 54, 147-154 (1989; Zbl 693.10040)] showed that a bounded number of summands for *A* is enough.

**Reviewer: ** A.Balog

**Classif.: ** * 11B13 Additive bases

11P55 Appl. of the Hardy-Littlewood method

11P99 Additive number theory

11B05 Topology etc. of sets of numbers

**Keywords: ** Hardy-Littlewood method

**Citations: ** Zbl 693.10040

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