Concentration of Points on Two and Three Dimensional Modular Hyperbolas and Applications

By Javier Cilleruelo, Moubariz Z. Garaev. 2010

68 citations recorded by the source corpus.

AbstractLet p be a large prime number, K, L, M, λ be integers with 1 ≤ M ≤ p and gcd(λ, p) = 1. The aim of our paper is to obtain sharp upper bound estimates for the number I2(M; K, L) of solutions of the congruence $$xy \equiv \lambda \quad ({\rm mod} p), \quad K+ 1 \leq x \leq K +M,\quad L+ 1 \leq y \leq L +M,$$and for the number I3(M;L) of solutions of the congruence $$xyz \equiv \lambda \quad ({\rm mod} p), \quad L+ 1 \leq x, y, z \leq L +M .$$Using the idea of Heath-Brown from [H], we obtain a bound for I2(M; K, L), which improves several recent results of Chan and Shparlinski [CS]. For instance, we prove that if M < p1/4, then I2(M; K, L) ≤ Mo(1).The problem with I3(M; L) is more difficult and requires a different approach. Here, we connect this problem with the Pell diophantine equation and prove that for M < p1/8 one has I3(M;L) ≤ Mo(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 $${{\mathbb{F}_{p}^{*}}}$$ of length $${{|\mathcal{I}_{1}| < p^{1/8}}}$$, then $$|\mathcal{I}_{1}\,\cdot \,\mathcal{I}_{2}\,\cdot \mathcal{I}_{3}| = (|\mathcal{I}_{1}|\,\cdot\,|\mathcal{I}_{2}|\,\cdot\,|\mathcal{I}_{3}|)^{1-o(1)}$$.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/4390157a9c0d0273d2c77c8cd7531b0b966f625e

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Citations in the corpus

On congruences with products of variables from short intervals and applications (2012, background): “Recently, motivated by some applications to certain algorithmic problems, new results on Jν(p, h, s;λ) have been given by Bourgain, Garaev, 1 Konyagin and Shparlinski [4].”
Modular hyperbolas (2011, background): “Upper bounds on the number of solutions of the congruence (43) with variables from short intervals are given in [36, 51].”
On the concentration of points of polynomial maps and applications (2012, methodology): “Here we use the approach of [5], combined with estimates on the Vinogradov mean value theorem (see [10, 11, 18] to estimate N(H; K, L).”
Growth Estimates in Positive Characteristic via Collisions (2015, result): “This may be viewed as an “almost-interval” generalisation of the much stronger results of Cilleruelo and Garaev [8] for intervals.”