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On the sum product estimates and two variables expanders. (English) Zbl 1219.11037

Let \(\mathbb{F}_p\) be the finite field of a prime order \(p\). Let \(F:\mathbb{F}_p\times \mathbb{F}_p\rightarrow \mathbb{F}_p\) be a function defined by \(F(x,y)=x(f(x)+by)\), where \(b\in \mathbb{F}_p^*\) and \(f:\mathbb{F}_p\rightarrow \mathbb{F}_p\) is any function. The author proves 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 \(b=1\), one can get the well-known sum-product theorem by J. Bourgain, N. Katz, and T. Tao [Geom. Funct. Anal. 14, No. 1, 27–57 (2004; Zbl 1145.11306)], and J. Bourgain, A. A. Glibichuk, and S. V. Konyagin [J. Lond. Math. Soc., II. Ser. 73, No. 2, 380–398 (2006; Zbl 1093.11057)], and also improves the previous known exponent from \(\frac{14}{13}\) to \(\frac{13}{12}\).

MSC:

11B30 Arithmetic combinatorics; higher degree uniformity
11B75 Other combinatorial number theory
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References:

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