Best Diophantine Approximations and Multidimensional Three Distance Theorem
Anton Shutov
TL;DR
This work builds a bridge between Diophantine approximation and multidimensional Three Distance phenomena via Chevallier's lemma, showing how bounds for the multidimensional g_dist statistic can be inferred from properties of best Diophantine approximations and lattice considerations. It establishes universal upper bounds such as $g_{\infty}(\alpha)\le 2^d+1$ and, in 2D, $g_2(\alpha)\le 5$, and analyzes the liminf behavior, obtaining $\underline{g}_{\infty}(\alpha)\le 2^d$ and, for almost all $\alpha$, $\underline{g}_{\infty}(\alpha)=1$. In 1D, it gives a complete dichotomy: $\overline{g}(\alpha)=3$ when $a_n=1$ infinitely often and $2$ otherwise, with the golden ratio case yielding $\underline{g}(\alpha)=2$ and the generic case $\underline{g}(\alpha)=1$. The paper also investigates the inverse direction, showing how large values of $g_{dist}$ force constraints on the growth of best approximations and proving almost-everywhere results such as $q_{n+2^{d-1}-1}<2q_n$ infinitely often in the $L^{\infty}$ norm.
Abstract
In 1996 N. Chevallier proved a beautiful lemma which connects Diophantine approximation and multidimensional generalizations of the famous Three Distance Theorem. Using this lemma we show how known results about multidimensional three distance theorem can be deduced from certain known results dealing with the best Diophantine approximations. Also we obtain some new results about liminf version of the problem. Beside this, we discuss the inverse problem: how results about multidimensional three distance theorem can be applied to study best Diophantine approximations.