The dispersion of dilated lacunary sequences, with applications in multiplicative Diophantine approximation
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| Format: | Preprint |
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2024
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| _version_ | 1866909233030627328 |
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| author | Stefanescu, Eduard |
| author_facet | Stefanescu, Eduard |
| contents | Let $(a_n)_{n \in \mathbb{N}}$ be a Hadamard lacunary sequence. We give upper bounds for the maximal gap of the set of dilates $\{a_n α\}_{n \leq N}$ modulo 1, in terms of $N$. For any lacunary sequence $(a_n)_{n \in \mathbb{N}}$ we prove the existence of a dilation factor $α$ such that the maximal gap is of order at most $(\log N)/N$, and we prove that for Lebesgue almost all $α$ the maximal gap is of order at most $(\log N)^{2+\varepsilon}/N$. The metric result is generalized to other measures satisfying a certain Fourier decay assumption. Both upper bounds are optimal up to a factor of logarithmic order, and the latter result improves a recent result of Chow and Technau. Finally, we show that our result implies an improved upper bound in the inhomogeneous version of Littlewood's problem in multiplicative Diophantine approximation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_19802 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The dispersion of dilated lacunary sequences, with applications in multiplicative Diophantine approximation Stefanescu, Eduard Number Theory 11J83, 11J70, 42A16, 28A78 Let $(a_n)_{n \in \mathbb{N}}$ be a Hadamard lacunary sequence. We give upper bounds for the maximal gap of the set of dilates $\{a_n α\}_{n \leq N}$ modulo 1, in terms of $N$. For any lacunary sequence $(a_n)_{n \in \mathbb{N}}$ we prove the existence of a dilation factor $α$ such that the maximal gap is of order at most $(\log N)/N$, and we prove that for Lebesgue almost all $α$ the maximal gap is of order at most $(\log N)^{2+\varepsilon}/N$. The metric result is generalized to other measures satisfying a certain Fourier decay assumption. Both upper bounds are optimal up to a factor of logarithmic order, and the latter result improves a recent result of Chow and Technau. Finally, we show that our result implies an improved upper bound in the inhomogeneous version of Littlewood's problem in multiplicative Diophantine approximation. |
| title | The dispersion of dilated lacunary sequences, with applications in multiplicative Diophantine approximation |
| topic | Number Theory 11J83, 11J70, 42A16, 28A78 |
| url | https://arxiv.org/abs/2406.19802 |