Rational Points in Translations of The Cantor Set

Abstract

Given two coprime integers $p\ge 2$ and $q\ge 3$, let $D_p\subset [0,1)$ consist of all rational numbers which have a finite $p$-ary expansion, and let $$K(q,\mathcal{A})=\{\sum _{i=1}^{\infty }\frac{d_i}{q^i}:d_i\in \mathcal{A}\forall i\in \mathbb{N}\},$$ where $\mathcal{A}\subset \{0,1,\dots ,q-1\}$ with cardinality $1<\#\mathcal{A}<q$. In 2021 Schleischitz showed that $\#(D_p\cap K(q,\mathcal{A}))<+\infty$. In this paper we show that for any $r\in \mathbb{Q}$ and for any $\alpha \in \mathbb{R}$, $$\#((r{D}_p+\alpha )\cap K(q,\mathcal{A}))<+\infty .$$

8 pages, final version