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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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