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On the distribution modulo 1 of exponential sequences. (English. Russian original) Zbl 1196.11107
Math. Notes 76, No. 2, 153-160 (2004); translation from Mat. Zametki 76, No. 2, 163-171 (2004).
Author’s summary: “New quantitative results on the intersection of winning sets and the Hausdorff dimension $$\dim$$ of this intersection are obtained. An application to the problem on fractional parts of the sequence $$\{ 2^n 3^m \alpha \}$$ is given.”
The author’s main result is Theorem 4. Let $$\| \cdot\|$$ denotes the distance to the nearest integer, $$(v_j)_{j=1}^\infty$$ be a sequence of positive real numbers such that $$\sum_{j=1}^\infty v_j^{-1} \leq 1/2$$, let $$(q_j)_{j=1}^\infty$$ be a sequence of integers such that $$q_j \geq 2$$, let $$(a_j)_{j=1}^\infty$$ be a sequence of real numbers such that $$a_j \geq 1$$, let $$(b_j)_{j=1}^\infty$$ be an arbitrary sequence of real numbers, and let $\kappa\geq \kappa_0 = \log(2^{18}3^2) = 14.67387383\dots .$ Then there is a set $$P_\kappa$$ of real numbers $$x$$ such that $$\forall x \in P_\kappa$$, $$\forall n \in\mathbb N_0$$, $$\forall j\in\mathbb N$$ one has the bound $\| q_j^n(a_jx + b_j)\| \geq \exp(-\kappa v_j) \quad\text{and}\quad \dim(P_\kappa) \geq 1-\kappa_0/\kappa.$
The author remarks that if $$\kappa\to\infty$$, then $$\dim(P_\kappa) \to 1$$.
Theorems 1 and 2 are corollaries of Theorem 4. Theorem 1 is obtained if in the conditions of Theorem 4 one takes $$q_j = 2$$, $$a_j = 3^j$$, $$b_j = 0$$ $$\forall j \in\mathbb N$$. Theorem 2 is obtained if in the conditions of Theorem 4 one takes $$q_j = j + 1$$, $$a_j = 1$$, $$b_j = 0$$ $$\forall j \in\mathbb N$$.

MSC:
 11K06 General theory of distribution modulo $$1$$ 11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
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