On the p-adic denseness of the quotient set of a polynomial image

Abstract

The quotient set, or ratio set, of a set of integers $A$ is defined as $R(A):=\left\{a/b:a,b\in A,b\ne 0\right\}$. We consider the case in which $A$ is the image of ${\mathbb{Z}}^{+}$ under a polynomial $f\in \mathbb{Z}[X]$, and we give some conditions under which $R(A)$ is dense in ${\mathbb{Q}}_p$. Then, we apply these results to determine when $R(S_m^n)$ is dense in ${\mathbb{Q}}_p$, where $S_m^n$ is the set of numbers of the form $x_1^n+\cdots +x_m^n$, with $x_1,\dots ,x_m\ge 0$ integers. This allows us to answer a question posed in [Garcia et al., $p$-adic quotient sets, Acta Arith. 179, 163-184]. We end leaving an open question.

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