## 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

### Keywords:

Sum-product estimates; expanders

### Citations:

Zbl 1145.11306; Zbl 1093.11057
Full Text:

### References:

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