Geodesic
Some results on Poincaré sets
Czechoslovak Mathematical Journal, Tome 70 (2020) no. 3, pp. 891-903
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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 $$ X_{H}:=\biggl \{ x=\sum ^{\infty }_{n=1} \frac {x_{n}}{2^{n}} \colon x_{n}\in \{0,1\}, x_{n} x_{n+h}=0 \ \text {for all} \ n\geq 1, \ h\in H\biggr \} $$ 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.
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 $$ X_{H}:=\biggl \{ x=\sum ^{\infty }_{n=1} \frac {x_{n}}{2^{n}} \colon x_{n}\in \{0,1\}, x_{n} x_{n+h}=0 \ \text {for all} \ n\geq 1, \ h\in H\biggr \} $$ 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.
DOI : 10.21136/CMJ.2020.0001-19
Classification : 11A07, 37B20
Keywords: Poincaré set; homogeneous set; Hausdorff dimension 
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Tang, Min-wei; Wu, Zhi-Yi. Some results on Poincaré sets. Czechoslovak Mathematical Journal, Tome 70 (2020) no. 3, pp. 891-903. doi: 10.21136/CMJ.2020.0001-19

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