Congruences involving product of intervals and sets with small multiplicative doubling modulo a prime and applications

In this paper we obtain new upper bound estimates for the number of solutions of the congruence $$\begin{equation} x\equiv y r\pmod p;\quad x,y\in \mathbb{N},\quad x,y\le H,\quad r\in \mathcal{U}, \end{equation}$$ for certain ranges of H and |${\mathcal U}$|, where ${\mathcal U}$ is a subset of the...

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Bibliographic Details
Published in:Mathematical proceedings of the Cambridge Philosophical Society 2016-05, Vol.160 (3), p.477-494
Main Authors: CILLERUELO, J., GARAEV, M. Z.
Format: Article
Language:English
Subjects:
Congruences
Constants
Estimates
Integers
Mathematical analysis
Mathematics
Polynomials
Prime numbers
Residues
Roots
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Summary:In this paper we obtain new upper bound estimates for the number of solutions of the congruence $$\begin{equation} x\equiv y r\pmod p;\quad x,y\in \mathbb{N},\quad x,y\le H,\quad r\in \mathcal{U}, \end{equation}$$ for certain ranges of H and |${\mathcal U}$|, where ${\mathcal U}$ is a subset of the field of residue classes modulo p having small multiplicative doubling. We then use these estimates to show that the number of solutions of the congruence $$\begin{equation} x^n\equiv \lambda\pmod p; \quad x\in \mathbb{N}, \quad L
ISSN:0305-0041
1469-8064
DOI:10.1017/S0305004115000808