Vol. 2, No. 5, 2009

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Some results on the size of sum and product sets of finite sets of real numbers

Derrick Hart and Alexander Niziolek

Vol. 2 (2009), No. 5, 603–609

Let A and B be finite subsets of positive real numbers. Solymosi gave the sum-product estimate max(|A + A|,|A A|) (4log|A|)13|A|43, where is the ceiling function. We use a variant of his argument to give the bound

max(|A + B|,|A B|) (4log|A|log|B|)13|A|23|B|23.

(This isn’t quite a generalization since the logarithmic losses are worse here than in Solymosi’s bound.)

Suppose that A is a finite subset of real numbers. We show that there exists an a A such that |aA + A| c|A|43 for some absolute constant c.

sum-product estimate, multiplicative energy, Solymosi bound
Mathematical Subject Classification 2000
Primary: 11B13, 11B75
Received: 7 September 2009
Accepted: 12 November 2009
Published: 13 January 2010

Communicated by Andrew Granville
Derrick Hart
Department of Mathematics
Rutgers University
Piscataway, NJ 08854-8019
United States
Alexander Niziolek
School of Engineering
Rutgers University
Piscataway, NJ 08854-8019
United States