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- M. Z. Garaev
- 2005

We remove logarithmic factors in error term estimates in asymp-totic formulas for the number of solutions of a class of additive con-gruences modulo a prime number.

- M. Z. Garaev
- 2007

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)

- M. Z. GARAEV
- 2007

Let F p be the field of prime order p. It is known that for any integer N ∈ [1, p] one can construct a subset A ⊂ F p with |A| = N such that max{|A + A|, |AA|} p 1/2 |A| 1/2. One of the results of the present paper implies that if A ⊂ F p with |A| > p 2/3 , then max{|A + A|, |AA|} p 1/2 |A| 1/2 .

We give upper bound estimates for the number of solutions of a certain diophantine equation. Our results can be applied to obtain new lower bound estimates for the L1-norm of certain exponential sums.

- M.Z. Garaev, F. Luca, I.E. Shparlinski, A. Winterhof
- IEEE Transactions on Information Theory
- 2006

For a Sidelnikov sequence of period p/sup m/-1, tight lower bounds are obtained on its linear complexity L over F/sub p/. In particular, these bounds imply that, uniformly over all p and m, L is close to its largest possible value p/sup m/-1.

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)

- M. Z. Garaev
- 2008

Let ε be a fixed positive quantity, m be a large integer, x j denote integer variables. We prove that for any positive integers N

- M. Z. GARAEV
- 2002

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)

- M. Z. Garaev
- Electr. J. Comb.
- 2008

Let F p be the field of residue classes modulo a prime number p. In this paper we prove that if A, B ⊂ F * p , then for any fixed ε > 0, |A + A| + |AB| min |B|, p |A| 1/25−ε |A|. This quantifies Bourgain's recent sum-product estimate.

- M. Z. Garaev
- 2005

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)