Some results on Poincaré sets

Min-wei Tang; Zhi-Yi Wu

Czechoslovak Mathematical Journal (2020)

  • Volume: 70, Issue: 3, page 891-903
  • ISSN: 0011-4642

Abstract

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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 ℋ ( X H ) = 0 , where X H : = x = ∑ n = 1 ∞ x n 2 n : x n ∈ { 0 , 1 } , x n x n + h = 0 for all n ≥ 1 , h ∈ H and dim 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.

How to cite

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Tang, Min-wei, and Wu, Zhi-Yi. "Some results on Poincaré sets." Czechoslovak Mathematical Journal 70.3 (2020): 891-903. <http://eudml.org/doc/297222>.

@article{Tang2020,
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 \[ X\_\{H\}:=\biggl \lbrace x=\sum ^\{\infty \}\_\{n=1\} \frac\{x\_\{n\}\}\{2^\{n\}\} \colon x\_\{n\}\in \lbrace 0,1\rbrace , x\_\{n\} x\_\{n+h\}=0 \ \text\{for all\} \ n\ge 1, \ h\in H\biggr \rbrace \] 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.},
author = {Tang, Min-wei, Wu, Zhi-Yi},
journal = {Czechoslovak Mathematical Journal},
keywords = {Poincaré set; homogeneous set; Hausdorff dimension},
language = {eng},
number = {3},
pages = {891-903},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Some results on Poincaré sets},
url = {http://eudml.org/doc/297222},
volume = {70},
year = {2020},
}

TY - JOUR
AU - Tang, Min-wei
AU - Wu, Zhi-Yi
TI - Some results on Poincaré sets
JO - Czechoslovak Mathematical Journal
PY - 2020
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 70
IS - 3
SP - 891
EP - 903
AB - 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 \[ X_{H}:=\biggl \lbrace x=\sum ^{\infty }_{n=1} \frac{x_{n}}{2^{n}} \colon x_{n}\in \lbrace 0,1\rbrace , x_{n} x_{n+h}=0 \ \text{for all} \ n\ge 1, \ h\in H\biggr \rbrace \] 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.
LA - eng
KW - Poincaré set; homogeneous set; Hausdorff dimension
UR - http://eudml.org/doc/297222
ER -

References

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