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 $$ K(q, \mathcal{A})=\bigg{ \sum_{i=1}^\infty \frac{d_i}{q^i}: d_i\in \mathcal{A}~ \forall i\in\mathbb{N} \bigg}, $$ where $\mathcal{A} \subset {0,1,\ldots, q-1}$ 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 $\alpha\in\mathbb{R}$, $$ #\big((r D_p+\alpha)\cap K(q,\mathcal{A})\big)<+\infty. $$