Circle coverings driven by arithmetic sequences: a percolation approach to Diophantine approximation and fractal intersections
Year: 2026
Type: preprint
Abstract: 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\|
Source: arXiv (Cornell University)
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7148714903
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