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Product of subsets of small intervals and points on exponential curves modulo a prime

Volume 193 / 2020

C. A. Díaz, M. Z. Garaev, J. Hernández Acta Arithmetica 193 (2020), 309-319 MSC: 11B50, 11D79, 11P21. DOI: 10.4064/aa181127-10-8 Published online: 27 January 2020

Abstract

Let $p$ be a large prime number, $h \gt 0$ and $s$ be integers, and $\mathcal {X}\subseteq [1,h]\cap \mathbb {Z}$. Following the work of Bourgain et al. (2013), we obtain nontrivial upper bounds for the number of solutions to the congruence $$ \prod \limits _{i=1}^4(x_i+s)\equiv \prod \limits _{j=1}^4(y_j+s)\not \equiv 0 \pmod {p},\ \quad x_i,y_j\in \mathcal {X}. $$ We apply these bounds to obtain new results on the number of integer points on exponential curves modulo a prime.

Authors

  • C. A. DíazCentro de Ciencias Matemáticas
    Universidad Nacional Autónoma de México
    Morelia 58089
    Michoacán, México
    e-mail
  • M. Z. GaraevCentro de Ciencias Matemáticas
    Universidad Nacional Autónoma de México
    Morelia 58089
    Michoacán, México
    e-mail
  • J. HernándezCentro de Ciencias Matemáticas
    Universidad Nacional Autónoma de México
    Morelia 58089
    Michoacán, México
    e-mail

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