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Let F p be the field of residue classes modulo a prime number p and let A be a non-empty subset of F p. In this paper we give an explicit version of the sum-product estimate of Bourgain, Katz, Tao and Bourgain, Glibichuk, Konyagin on the size of max{|A + A|, |AA|}. In particular, our result implies that if 1 < |A| ≤ p 7/13 (log p) −4/13 , then max{|A + A|,(More)
In the present paper we obtain new upper bound estimates for the number of solutions of the congruence x ≡ yr (mod p); x, y ∈ N, x, y ≤ H, r ∈ U , for certain ranges of H and |U|, where U is a subset of the field of residue classes modulo p having small multiplicative doubling. We then use this estimate to show that the number of solutions of the congruence(More)
We study the sets {g x −g y (mod p) : 1 ≤ x, y ≤ N} and {xy : 1 ≤ x, y ≤ N} where p is a large prime number, g is a primitive root, and p 2/3 < N < p. 1. Introduction. Let p be a large prime number, g a primitive root (mod p), and N a given positive integer, N < p. In a series of papers, the distribution of powers g n (mod p) has been investigated by [1, 2,(More)
For a large integer m, we obtain an asymptotic formula for the number of solutions of a certain congruence modulo m with four variables , where the variables belong to special sets of residue classes mod-ulo m. This formula are applied to obtain a new bound for a double trigonometric sum with an exponential function and new information on the exceptional(More)