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.

中文翻译：

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Poincaré 集的一些结果

已知正整数集 H 是庞加莱集（也称为交集，参见 I. Ruzsa (1982)）当且仅当 $${\dim _\mathcal{H}}({X_H}) = 0$$ , 其中 $${X_H}: = \left\{ {x = \sum\limits_{n = 1}^\infty {\frac{{{x_n}}}{{{2^n}}} :{x_n} \in \{ 0,1\} ,{x_n}{x_n} + h = 0\,for\,all\,n \geqslant 1,\,h \in H} } \right\}$ $ 和 $${\dim _\mathcal{H}}$$ 表示 Hausdorff 维度（参见 C. Bishop, Y. Peres (2017), Theorem 2.5.5）。在本文中，我们通过将 2 替换为 b > 2 来研究集合 XH。令人惊讶的是，有一些新的现象值得研究。我们研究它们并给出几个例子来解释我们的结果。