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 \(\frac{13}{12}\).


11B30 Arithmetic combinatorics; higher degree uniformity
11B75 Other combinatorial number theory
Full Text: DOI Euclid


[1] J. Bourgain, More on the sum-product phenomenon in prime fields and its applications, Int. J. Number Theory 1(1) (2005), 1\Ndash32. · Zbl 1173.11310
[2] J. Bourgain and M. Z. Garaev, On a variant of sum-product estimates and explicit exponential sum bounds in prime fields, Math. Proc. Cambridge Philos. Soc. 146(1) (2009), 1\Ndash21. · Zbl 1194.11086
[3] J. Bourgain, A. A. Glibichuk, and S. V. Konyagin, Estimates for the number of sums and products and for exponential sums in fields of prime order, J. London Math. Soc. (2) 73(2) (2006), 380\Ndash398. · Zbl 1093.11057
[4] J. Bourgain, N. Katz, and T. Tao, A sum-product estimate in finite fields, and applications, Geom. Funct. Anal. 14(1) (2004), 27\Ndash57. · Zbl 1145.11306
[5] M. Z. Garaev, An explicit sum-product estimate in \(\mathbb{F}_ p\), Int. Math. Res. Not. IMRN 11 (2007), Art. ID rnm035, 11 pp. · Zbl 1160.11014
[6] M. Z. Garaev, The sum-product estimate for large subsets of prime fields, Proc. Amer. Math. Soc. 136(8) (2008), 2735\Ndash2739. · Zbl 1163.11017
[7] M. Z. Garaev and C.-Y. Shen, On the size of the set \(A(A+1)\), Math. Z. (2009), · Zbl 1237.11004
[8] A. A. Glibichuk and S. V. Konyagin, Additive properties of product sets in fields of prime order, in: “Additive combinatorics” , CRM Proc. Lecture Notes 43 , Amer. Math. Soc., Providence, RI, 2007, pp. 279\Ndash286. · Zbl 1215.11020
[9] N. H. Katz and C.-Y. Shen, A slight improvement to Garaev’s sum product estimate, Proc. Amer. Math. Soc. 136(7) (2008), 2499\Ndash2504. · Zbl 1220.11013
[10] N. H. Katz and C.-Y. Shen, Garaev’s inequality in finite fields not of prime order, Online J. Anal. Comb. 3 (2008), 6 pp. · Zbl 1241.11023
[11] J. Solymosi, An upper bound on the multiplicative energy,
[12] T. Tao and V. Vu, “Additive combinatorics” , Cambridge Studies in Advanced Mathematics 105 , Cambridge University Press, Cambridge, 2006. · Zbl 1127.11002
[13] V. H. Vu, Sum-product estimates via directed expanders, Math. Res. Lett. 15(2) (2008), 375\Ndash388. · Zbl 1214.11021
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