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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 AA is defined as R(A):={a/b:a,bA,  b0}R(A) := \left\{a/b : a,b \in A,\; b \neq 0\right\}. We consider the case in which AA is the image of Z+\mathbb{Z}^+ under a polynomial fZ[X]f \in \mathbb{Z}[X], and we give some conditions under which R(A)R(A) is dense in Qp\mathbb{Q}_p. Then, we apply these results to determine when R(Smn)R(S_m^n) is dense in Qp\mathbb{Q}_p, where SmnS_m^n is the set of numbers of the form x1n++xmnx_1^n + \cdots + x_m^n, with x1,,xm0x_1, \dots, x_m \geq 0 integers. This allows us to answer a question posed in [Garcia et al., pp-adic quotient sets, Acta Arith. 179, 163-184]. We end leaving an open question

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