$p$-adic quotient sets

Stephan Ramon Garcia¹ ; Yu Xuan Hong² ; Florian Luca³ ; Elena Pinsker² ; Carlo Sanna⁴ ; Evan Schechter² ; Adam Starr²

¹ Department of Mathematics Pomona College 610 N. College Ave. Claremont, CA 91711, U.S.A. <a href="http://pages.pomona.edu/~sg064747">http://pages.pomona.edu/~sg064747</a>
² Department of Mathematics Pomona College 610 N. College Ave. Claremont, CA 91711
³ School 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
⁴ Department of Mathematics Università degli Studi di Torino Via Carlo Alberto, 10 10123 Torino, Italy

Acta Arithmetica, Tome 179 (2017) no. 2, pp. 163-184.

Résumé

For $A\subseteq \mathbb{N}$, the question of when $R(A)=\{a/a^{\prime }:a,a^{\prime }\in A\}$ 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.

Détail

DOI : 10.4064/aa8579-4-2017

Mots-clés : subseteq mathbb question dense positive real numbers mathbb has examined many authors years contrast p adic setting largely unexplored investigate conditions under which dense p adic numbers techniques elementary algebraic analytic number theory employed pose many questions should general interest