Circle coverings driven by arithmetic sequences: a percolation approach to Diophantine approximation and fractal intersections
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arXiv
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| Natura: | Preprint |
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2026
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| author | Hauke, Manuel Shubin, Andrei Stefanescu, Eduard Zafeiropoulos, Agamemnon |
| author_facet | Hauke, Manuel Shubin, Andrei Stefanescu, Eduard Zafeiropoulos, Agamemnon |
| contents | We study problems on covering $[0,1)$ by shrinking intervals centered at the points $\{q_n x\}$, where $(q_n)_{n\in \mathbb{N}}$ is a given real-valued sequence and $x \in [0,1)$ is random. For real-valued lacunary sequences $(q_n)_{n\in\mathbb{N}}$, we show that the covering radius $\frac{1}{n}$ is sharp up to a constant: there exist $C>c>0$ such that, for Lebesgue-almost all $x$, the intervals of length $\frac{C}{n}$ cover $[0,1)$ infinitely often, while this fails for intervals of length $\frac{c}{n}$. Moreover, the lower bound holds for certain sub-lacunary rates and the results partially extend to all probability measures with sufficiently fast Fourier decay. As an application, we obtain a new bound for a variant of the inhomogeneous Littlewood-Cassels problem: for any badly approximable $α$ and $γ\in\mathbb{R}$, there exists a set of badly approximable $β$ of full Hausdorff dimension such that
$ \|nα-γ\| \|nβ-δ\|<C/(n\log n)$ for infinitely many $n\geqslant 1,$ uniformly in $δ\in\mathbb{R}$. This improves upon previous works of Haynes-Jensen-Kristensen, Chow-Technau, and the third author, and is best possible when one restricts to best approximations of the first factor. Second, under certain arithmetic restrictions on $(q_n)_{n\in\mathbb{N}}$, we compute the almost-sure Hausdorff dimension of limsup sets generated by intervals of size $\frac{1}{n^ν}$ for $ν\geqslant 1$, centered at $\{q_n x\}$, and intersected with Ahlfors regular compact sets such as the middle-third Cantor set. In particular, our results apply to all real-valued lacunary sequences, to integer-valued polynomials, and to powers of primes. This substantially extends the work of Bugeaud and Durand, which applies only to certain super-lacunary integer-valued sequences. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_02005 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Circle coverings driven by arithmetic sequences: a percolation approach to Diophantine approximation and fractal intersections Hauke, Manuel Shubin, Andrei Stefanescu, Eduard Zafeiropoulos, Agamemnon Number Theory Dynamical Systems Probability We study problems on covering $[0,1)$ by shrinking intervals centered at the points $\{q_n x\}$, where $(q_n)_{n\in \mathbb{N}}$ is a given real-valued sequence and $x \in [0,1)$ is random. For real-valued lacunary sequences $(q_n)_{n\in\mathbb{N}}$, we show that the covering radius $\frac{1}{n}$ is sharp up to a constant: there exist $C>c>0$ such that, for Lebesgue-almost all $x$, the intervals of length $\frac{C}{n}$ cover $[0,1)$ infinitely often, while this fails for intervals of length $\frac{c}{n}$. Moreover, the lower bound holds for certain sub-lacunary rates and the results partially extend to all probability measures with sufficiently fast Fourier decay. As an application, we obtain a new bound for a variant of the inhomogeneous Littlewood-Cassels problem: for any badly approximable $α$ and $γ\in\mathbb{R}$, there exists a set of badly approximable $β$ of full Hausdorff dimension such that $ \|nα-γ\| \|nβ-δ\|<C/(n\log n)$ for infinitely many $n\geqslant 1,$ uniformly in $δ\in\mathbb{R}$. This improves upon previous works of Haynes-Jensen-Kristensen, Chow-Technau, and the third author, and is best possible when one restricts to best approximations of the first factor. Second, under certain arithmetic restrictions on $(q_n)_{n\in\mathbb{N}}$, we compute the almost-sure Hausdorff dimension of limsup sets generated by intervals of size $\frac{1}{n^ν}$ for $ν\geqslant 1$, centered at $\{q_n x\}$, and intersected with Ahlfors regular compact sets such as the middle-third Cantor set. In particular, our results apply to all real-valued lacunary sequences, to integer-valued polynomials, and to powers of primes. This substantially extends the work of Bugeaud and Durand, which applies only to certain super-lacunary integer-valued sequences. |
| title | Circle coverings driven by arithmetic sequences: a percolation approach to Diophantine approximation and fractal intersections |
| topic | Number Theory Dynamical Systems Probability |
| url | https://arxiv.org/abs/2604.02005 |