Some Results on Poincaré Sets

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

$${X_H}: = \left\{ {x = \sum\limits_{n = 1}^\infty {\frac{{{x_n}}}{{{2^n}}}:{x_n} \in \{ 0,1\} ,{x_n}{x_n} + h = 0\,for\,all\,n \geqslant 1,\,h \in H} } \right\}$$

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.