Czechoslovak Mathematical Journal, Vol. 70, No. 3, pp. 891-903, 2020
Some results on Poincaré sets
Min-wei Tang, Zhi-Yi Wu
Received January 1, 2019. Published online March 27, 2020.
Abstract: It is known that a set $H$ of positive integers is a Poincaré set (also called intersective set, see I. Ruzsa (1982)) if and only if $\dim_{\mathcal{H}}(X_H)=0$, where $X_H:=\biggl\{ x=\sum^{\infty}_{n=1} \frac{x_n}{2^n} \colon x_n\in\{0,1\},\ x_n x_{n+h}=0\ \text{for all} \ n\geq1, \ h\in H\biggr\}$ and $\dim_{\mathcal{H}}$ denotes the Hausdorff dimension (see C. Bishop, Y. Peres (2017), Theorem 2.5.5). In this paper we study the set $X_H$ by replacing $2$ with $b>2$. It is surprising that there are some new phenomena to be worthy of studying. We study them and give several examples to explain our results.
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Affiliations: Min-wei Tang, School of Mathematics and Statistics, Wuhan University, 299# Ba Yi Road, Wuchang District, Wuhan, Hubei Province, 430072, P. R. China, e-mail: tmw33@163.com, Zhi-Yi Wu (corresponding author), School of Mathematics, Sun Yat-Sen University, No. 135, Xingang Xi Road, Guangzhou, 510275, P. R. China, e-mail: zhiyiwu@126.com
