Rational Points in Translations of The Cantor Set

TL;DR

This paper proves that the intersection of any rational translation of the set of rationals with finite p-ary expansion and a certain Cantor-like set is finite, extending previous results on the finiteness of such intersections.

Contribution

It generalizes prior finiteness results to include rational translations of the set, showing these intersections remain finite.

Findings

The intersection of rationally translated D_p with K(q, A) is finite.

This finiteness holds for any rational translation and any real shift.

Extends previous work on the intersection properties of these sets.

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 $\alpha\in\mathbb{R}$, $$ \#\big((r D_p+\alpha)\cap K(q,\mathcal{A})\big)<+\infty. $$