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

Abstract

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

$max ( | A + B | , | A ⋅ B | ) ≥ ( 4 ⌈ log | A | ⌉ ⌈ log | B | ⌉ ) − 1 ∕ 3 | A | 2 ∕ 3 | B | 2 ∕ 3 .$

(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 $a∈A$ such that $|aA+A|≥c|A|4∕3$ for some absolute constant $c$.

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.involve/1513799214

Digital Object Identifier
doi:10.2140/involve.2009.2.603

Mathematical Reviews number (MathSciNet)
MR2601580

Zentralblatt MATH identifier
1246.11024

Citation

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. https://projecteuclid.org/euclid.involve/1513799214

References

• N. Alon and J. H. Spencer, The probabilistic method, 2nd ed., Wiley-Interscience, New York, 2000.
• J. Bourgain, “Mordell's exponential sum estimate revisited”, J. Amer. Math. Soc. 18:2 (2005), 477–499.
• J. Chapman, M. B. Erdoğan, D. Hart, A. Iosevich, and D. Koh, “Pinned distance sets, $k$-simplices, Wolff's exponent in finite fields and sum-product estimates”, preprint, 2009.
• G. Elekes, “On the number of sums and products”, Acta Arith. 81:4 (1997), 365–367.
• P. Erdős and E. Szemerédi, “On sums and products of integers”, pp. 213–218 in Studies in pure mathematics, Birkhäuser, Basel, 1983.
• K. Ford, “Sums and products from a finite set of real numbers”, Ramanujan J. 2:1-2 (1998), 59–66.
• D. Hart and A. Iosevich, “Sums and products in finite fields: an integral geometric viewpoint”, pp. 129–135 in Radon transforms, geometry, and wavelets, Contemp. Math. 464, Amer. Math. Soc., Providence, RI, 2008.
• M. B. Nathanson, “On sums and products of integers”, Proc. Amer. Math. Soc. 125:1 (1997), 9–16.
• I. E. Shparlinski, “On the solvability of bilinear equations in finite fields”, Glasg. Math. J. 50:3 (2008), 523–529.
• J. Solymosi, “On the number of sums and products”, Bull. London Math. Soc. 37:4 (2005), 491–494.
• J. Solymosi, “Bounding multiplicative energy by the sumset”, Adv. Math. 222:2 (2009), 402–408.
• E. Szemerédi and W. T. Trotter, Jr., “Extremal problems in discrete geometry”, Combinatorica 3:3-4 (1983), 381–392.
• T. Tao and V. Vu, Additive combinatorics, vol. 105, Cambridge University Press, Cambridge, 2006.