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$p$-adic quotient sets

Volume 179 / 2017

Stephan Ramon Garcia, Yu Xuan Hong, Florian Luca, Elena Pinsker, Carlo Sanna, Evan Schechter, Adam Starr Acta Arithmetica 179 (2017), 163-184 MSC: Primary 11B05; Secondary 11A07, 11B39. DOI: 10.4064/aa8579-4-2017 Published online: 16 June 2017

Abstract

For $A \subseteq \mathbb{N}$, the question of when $R(A) = \{a/a’ : a, a’ \in A\}$ is dense in the positive real numbers $\mathbb{R}_+$ has been examined by many authors over the years. In contrast, the $p$-adic setting is largely unexplored. We investigate conditions under which $R(A)$ is dense in the $p$-adic numbers. Techniques from elementary, algebraic, and analytic number theory are employed. We also pose many open questions that should be of general interest.

Authors

  • Stephan Ramon GarciaDepartment of Mathematics
    Pomona College
    610 N. College Ave.
    Claremont, CA 91711, U.S.A.
    http://pages.pomona.edu/~sg064747
    e-mail
  • Yu Xuan HongDepartment of Mathematics
    Pomona College
    610 N. College Ave.
    Claremont, CA 91711
  • Florian LucaSchool of Mathematics
    University of the Witwatersrand
    Private Bag 3
    Wits 2050, Johannesburg, South Africa
    and
    Max Planck Institute for Mathematics
    Vivatsgasse 7
    53111 Bonn, Germany
    e-mail
  • Elena PinskerDepartment of Mathematics
    Pomona College
    610 N. College Ave.
    Claremont, CA 91711
  • Carlo SannaDepartment of Mathematics
    Università degli Studi di Torino
    Via Carlo Alberto, 10
    10123 Torino, Italy
    e-mail
  • Evan SchechterDepartment of Mathematics
    Pomona College
    610 N. College Ave.
    Claremont, CA 91711
  • Adam StarrDepartment of Mathematics
    Pomona College
    610 N. College Ave.
    Claremont, CA 91711

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