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

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.