Involve: A Journal of Mathematics

  • Involve
  • Volume 2, Number 5 (2009), 603-609.

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

Derrick Hart and Alexander Niziolek

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Let A and B be finite subsets of positive real numbers. Solymosi gave the sum-product estimate max(|A+A|,|AA|)(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 | ) ( 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 aA such that |aA+A|c|A|43 for some absolute constant c.

Article information

Involve, Volume 2, Number 5 (2009), 603-609.

Received: 7 September 2009
Accepted: 12 November 2009
First available in Project Euclid: 20 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11B13: Additive bases, including sumsets [See also 05B10] 11B75: Other combinatorial number theory

sum-product estimate multiplicative energy Solymosi bound


Hart, Derrick; Niziolek, Alexander. Some results on the size of sum and product sets of finite sets of real numbers. Involve 2 (2009), no. 5, 603--609. doi:10.2140/involve.2009.2.603.

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