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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