Open Access
2010 On the Sum Product Estimates and Two Variables Expanders
Chun-Yen Shen
Publ. Mat. 54(1): 149-157 (2010).


Let ${\mathbb F}_p$ be the finite field of a prime order $p$. Let $F \colon {\mathbb F}_p \times {\mathbb F}_p\rightarrow {\mathbb F}_p$ be a function defined by $F(x,y)=x(f(x)+by)$, where $b \in {\mathbb F}_p^*$ and $f\colon {\mathbb F}_p \rightarrow {\mathbb F}_p$ is any function. We prove that if $A \subset {\mathbb F}_p$ and $|A|<p^{1/2}$ then

$|A+A|+|F(A,A)| \gtrapprox |A|^{\frac{13}{12}}.$

Taking $f=0$ and $b=1$, we get the well-known sum-product theorem by Bourgain, Katz and Tao, and Bourgain, Glibichuk and Konyagin, and also improve the previous known exponent from $\frac{14}{13}$ to $\frac{13}{12}$.


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Chun-Yen Shen. "On the Sum Product Estimates and Two Variables Expanders." Publ. Mat. 54 (1) 149 - 157, 2010.


Published: 2010
First available in Project Euclid: 8 January 2010

zbMATH: 1219.11037
MathSciNet: MR2603593

Primary: 11B75

Keywords: Expanders , products , sums

Rights: Copyright © 2010 Universitat Autònoma de Barcelona, Departament de Matemàtiques

Vol.54 • No. 1 • 2010
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