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 $K(q,A)=\left(\sum _{i=1}^{\infty }\frac{d_i}{q^i}:d_i\in A\forall i\in N\right),$where $A\subset \left(0,1,\dots ,q-1\right)$ with cardinality $1<\#A<q$. In 2021 Schleischitz showed that $\#(D_p\cap K(q,A))<+\infty$. In this paper we show that for any $r\in Q$ and for any $\alpha \in R$, $\#\left((r{D}_p+\alpha )\cap K(q,A)\right)<+\infty .$