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Keywords:
Poincaré set; homogeneous set; Hausdorff dimension
Poincaré set; homogeneous set; Hausdorff dimension
Summary:
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.
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