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

TL;DR

This paper investigates conditions under which the ratio set of polynomial images of positive integers is dense in the p-adic numbers, applying results to sum of powers sets and addressing a question from prior research.

Contribution

It provides new criteria for p-adic density of quotient sets of polynomial images and applies these to sum of powers sets, extending understanding in p-adic number theory.

Findings

Identifies conditions for p-adic density of R(A) for polynomial images.

Determines density of R(S_m^n) in rdic fields for sum of powers sets.

Answers a previously posed question on p-adic quotient sets.

Abstract

The quotient set, or ratio set, of a set of integers $A$ is defined as $R(A) := \left\{a/b : a,b \in A,\; b \neq 0\right\}$. 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 $x_1, \dots, x_m \geq 0$ 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.