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

Hart, Derrick; Niziolek, Alexander

doi:10.2140/involve.2009.2.603

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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

DOI: 10.2140/involve.2009.2.603

Abstract

Let $A$ and $B$ be finite subsets of positive real numbers. Solymosi gave the sum-product estimate $max (| A + A |, | A \cdot A |) \geq {(4 ⌈ log | A | ⌉)}^{- 1 ∕ 3} | A |^{4 ∕ 3}$ , where $⌈ ⌉$ is the ceiling function. We use a variant of his argument to give the bound

max (| A + B |, | A \cdot B |) \geq {(4 ⌈ log | A | ⌉ ⌈ log | B | ⌉)}^{- 1 ∕ 3} | A |^{2 ∕ 3} | B |^{2 ∕ 3} .

(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 $a \in A$ such that $| a A + A | \geq c | A |^{4 ∕ 3}$ for some absolute constant $c$ .

Keywords

sum-product estimate, multiplicative energy, Solymosi bound

Mathematical Subject Classification 2000

Primary: 11B13, 11B75

Milestones

Received: 7 September 2009

Accepted: 12 November 2009

Published: 13 January 2010

Communicated by Andrew Granville

Authors

Derrick Hart
Department of Mathematics Rutgers University Piscataway, NJ 08854-8019 United States
http://www.math.rutgers.edu/~dnhart/
Alexander Niziolek
School of Engineering Rutgers University Piscataway, NJ 08854-8019 United States

Vol. 2, No. 5, 2009