Bounds on the gaps in the fractional parts of a linear form

TL;DR

This work extends the classical three-gap phenomenon to the setting of fractional parts of linear forms, bounding the sizes of gaps in $K(B;Q)$ for generic $B$ as $Q$ grows. By translating the problem to a lattice framework via $U_B$ and $A_Q$, the authors leverage geometry-of-numbers techniques and Kleinbock–Margulis diagonal-flow zero-one laws to derive upper and lower bounds on the gap function $\gamma_{(B;Q)}(\mathbf{v})$, with explicit dependence on an auxiliary function $f$ satisfying $\sum_{k} 1/f(k)<\infty$. They establish that for almost all $B$, the maximal and minimal gap scales obey bounds of the form $Q^{-1/m}$ times polylogarithmic factors, and, under a spherical cover condition, the gap scales as $\ll Q^{-1/m}$; these results also yield a Diophantine-approximation corollary with uniform congruence constraints. The methods combine a lattice reformulation, successive-minima analyses, and zero-one laws to connect gap-size bounds with Diophantine uniform-approximation phenomena, contributing new insights into gap distributions and their arithmetic implications.

Abstract

We provide bounds on the sizes of the gaps -- defined broadly -- in the set $\{k_1β_1 + \ldots + k_nβ_n \mbox{ (mod 1)} : k_i \in \mathbb Z \cap (0,Q^\frac{1}{n}]\}$ for generic $β_1, \ldots, β_n \in \mathbb R^m$ and all sufficiently large $Q$. We also introduce a related problem in Diophantine approximation, which we believe is of independent interest.

Theorems & Definitions (15)

• Theorem 1.1
• Corollary
• Theorem 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• proof : Proof of Theorem \ref{['thm:existence']}
• Lemma 2.4
• proof