62 research outputs found

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

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    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

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    Let pp be a large prime number, K,L,M,λK,L,M,\lambda be integers with 1Mp1\le M\le p and gcd(λ,p)=1.{\color{red}\gcd}(\lambda,p)=1. The aim of our paper is to obtain sharp upper bound estimates for the number I2(M;K,L)I_2(M; K,L) of solutions of the congruence xyλ(modp),K+1xK+M,L+1yL+M 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 I3(M;L)I_3(M;L) of solutions of the congruence xyzλ(modp),L+1x,y,zL+M.xyz\equiv\lambda\pmod p, \quad L+1\le x,y,z\le L+M. We obtain a bound for I2(M;K,L),I_2(M;K,L), which improves several recent results of Chan and Shparlinski. For instance, we prove that if M<p1/4,M<p^{1/4}, then I2(M;K,L)Mo(1).I_2(M;K,L)\le M^{o(1)}. For I3(M;L)I_3(M;L) we prove that if M<p1/8M<p^{1/8} then I3(M;L)Mo(1).I_3(M;L)\le M^{o(1)}. Our results have applications to some other problems as well. For instance, it follows that if I1,I2,I3\mathcal{I}_1, \mathcal{I}_2, \mathcal{I}_3 are intervals in \F^*_p of length Ii<p1/8,|\mathcal{I}_i|< p^{1/8}, then I1I2I3=(I1I2I3)1o(1). |\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
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