Circle coverings driven by arithmetic sequences: a percolation approach to Diophantine approximation and fractal intersections

TL;DR

This paper investigates how well sequences of shrinking intervals centered at arithmetic sequence points cover the unit interval, with applications to Diophantine approximation and fractal set intersections, extending previous results to broader classes of sequences.

Contribution

It establishes sharp bounds for covering intervals driven by lacunary sequences, extends Hausdorff dimension results to various sequences, and improves bounds in inhomogeneous Diophantine approximation problems.

Findings

Covering radius of 1/n is sharp up to a constant for lacunary sequences.

Almost-sure Hausdorff dimension of limsup sets is computed for various sequences.

Improves bounds in inhomogeneous Littlewood-Cassels problem for badly approximable numbers.

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 $\alpha$ and $\gamma\in\mathbb{R}$, there exists a set of badly approximable $\beta$ of full Hausdorff dimension such that $ \|n\alpha-\gamma\| \|n\beta-\delta\|<C/(n\log n)$ for infinitely many $n\geqslant 1,$ uniformly in $\delta\in\mathbb{R}$. 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 $\frac{1}{n^{\nu}}$ for $\nu \geqslant 1$, 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.