Rational points in translations of the Cantor set

Abstract

Given two coprime integers p≥2 and q≥3, let Dp⊂[0,1) consist of all rational numbers which have a finite p-ary expansion, and let K(q,A)=∑i=1∞diqi:di∈A∀i∈N,where A⊂0,1,…,q−1 with cardinality 1<#A<q. In Schleischitz (2021) showed that #(Dp∩K(q,A))<+∞. In this paper we show that for any r∈Q and for any α∈R, #((rDp+α)∩K(q,A))<+∞.