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arXiv · 1605.00311

Central limit theorems for simultaneous Diophantine approximations

Dmitry Dolgopyat·Bassam Fayad·Ilya Vinogradov
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Abstract

We study the distribution modulo $1$ of the values taken on the integers of $r$ linear forms in $d$ variables with random coefficients. We obtain quenched and annealed central limit theorems for the number of simultaneous hits into shrinking targets of radii $n^{-\frac{r}{d}}$. By the Khintchine-Groshev theorem on Diophantine approximations, $\frac{r}{d}$ is the critical exponent for the infinite number of hits.

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Dmitry Dolgopyat, Bassam Fayad, Ilya Vinogradov. 2016-05-01. Central limit theorems for simultaneous Diophantine approximations. https://arxiv.org/abs/1605.00311

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