An improved sum-product inequality in fields of prime order

This note improves the best known exponent 1/12 in the prime field sum-product inequality (for small sets) to 1/11, modulo a logarithmic factor.Comment: I spotted that the existing version was, in fact, not the final on

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

We remove logarithmic factors in error term estimates in asymptotic formulas for the number of solutions of a class of additive congruences modulo a prime number.Comment: 13 page

Concentration points on two and three dimensional modular hyperbolas and applications

Let $p$ be a large prime number, $K,L,M,\lambda$ be integers with $1\le M\le p$ and ${\color{red}\gcd}(\lambda,p)=1.$ The aim of our paper is to obtain sharp upper bound estimates for the number $I_2(M; K,L)$ of solutions of the congruence $$xy\equiv\lambda \pmod p, \qquad K+1\le x\le K+M,\quad L+1\le y\le L+M$$ and for the number $I_3(M;L)$ of solutions of the congruence $$xyz\equiv\lambda\pmod p, \quad L+1\le x,y,z\le L+M.$$ We obtain a bound for $I_2(M;K,L),$ which improves several recent results of Chan and Shparlinski. For instance, we prove that if $M<p^{1/4},$ then $I_2(M;K,L)\le M^{o(1)}.$ For $I_3(M;L)$ we prove that if $M<p^{1/8}$ then $I_3(M;L)\le M^{o(1)}.$ Our results have applications to some other problems as well. For instance, it follows that if $\mathcal{I}_1, \mathcal{I}_2, \mathcal{I}_3$ are intervals in \F^*_p of length $|\mathcal{I}_i|< p^{1/8},$ then $$|\mathcal{I}_1\cdot \mathcal{I}_2\cdot \mathcal{I}_3|= (|\mathcal{I}_1|\cdot |\mathcal{I}_2|\cdot |\mathcal{I}_3|)^{1-o(1)}.$$Comment: 12 page