Moubariz Z. Garaev

On a variant of sum-product estimates and explicit exponential sum bounds in prime fields

Let F p be the field of a prime order p and F * p be its multiplicative subgroup. In this paper we obtain a variant of sum-product estimates which in particular implies the bound for any subset A ⊂ F p with 1 A | p 12/23 . Then we apply our estimate to obtain explicit bounds for some exponential sums in F p . We show that for any subsets X , Y , Z ⊂ F * p and any complex numbers α x , β y , γ z with |α x | ≤ 1, |β y | ≤ 1, |γ z | ≤ 1, the following bound holds: We apply this bound further to show that if H is a subgroup of F * p with | H | > p 1/4 , then Finally we show that if g is a generator of F * p then for any M p the number of solutions of the equation is less than . This implies that if p 1/2 M p , then

The sum-product estimate for large subsets of prime fields

Let Fp be the field of prime order p. It is known that for any integer N ∈ [1,p] one can construct a subset A C Fp 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 C Fp with |A| > p 2/3 , then max{|A + A|, |AA|}» p 1/2 |A| 1/2 .

On congruences with products of variables from short intervals and applications

We obtain upper bounds on the number of solutions to congruences of the type (x1 + s)... (xν + s) ≡ (y1 + s)... (xν + s) ≢ 0 (mod p) modulo a prime p with variables from some short intervals. We give some applications of our results and in particular improve several recent estimates of J. Cilleruelo and M.Z. Garaev on exponential congruences and on cardinalities of products of short intervals, some double character sum estimates of J. Friedlander and H. Iwaniec and some results of M.-C. Chang and A.A. Karatsuba on character sums twisted with the divisor function.

On the Hidden Shifted Power Problem

We consider the problem of recovering a hidden element

On the logarithmic factor in error term estimates in certain additive congruence problems

s

DISTRIBUTION OF SPECIAL SEQUENCES MODULO A LARGE PRIME

of a finite field

Upper bounds for the number of solutions of a Diophantine equation

\F_q

Exponential sums and congruences with factorials

of

Density of non-residues in Burgess-type intervals and applications

q

THE REPRESENTATION OF RESIDUE CLASSES BY PRODUCTS OF SMALL INTEGERS

elements from queries to an oracle that for a given