论文标题
Cantor集的翻译中的理性点
Rational Points in Translations of The Cantor Set
论文作者
论文摘要
给定两个副本整数$ p \ ge 2 $和$ q \ ge 3 $，让$ d_p \ subset [0,1）$由所有具有有限$ p $ p $ - ary扩展的理性数字组成，然后$ $ k（q，\ nathcal {a} a} \ frac {d_i} {q^i}：d_i \ in \ mathcal {a}〜\ forall i \ in \ mathbb {n} \ bigG \}，$ \ mathcal {a a} a} a} $ 1 <\＃\ Mathcal {a} <q $。在2021年，schleischitz表明$ \＃（d_p \ cap k（q，\ mathcal {a}）））<+\ \ infty $。在本文中，我们表明，对于任何$ r \ in \ mathbb {q} $，对于任何$α\ in \ mathbb {r} $，$$ \＃\ big（（（r d_p+α）\ cap k（q，mathcal {a}） $$
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 $α\in\mathbb{R}$, $$ \#\big((r D_p+α)\cap K(q,\mathcal{A})\big)<+\infty. $$
