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 2021 Schleischitz 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))<+∞.
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