Equidistribution in the space of 3-lattices and Dirichlet-improvable vectors on planar lines

TL;DR

This paper characterizes when measures supported on lines in the space of 3-lattices become equidistributed under certain flows, linking Diophantine properties of the lines to number-theoretic approximation properties.

Contribution

It provides explicit Diophantine criteria for equidistribution of measures on the space of 3-lattices, connecting dynamics with Diophantine approximation on planar lines.

Findings

Measures become equidistributed if and only if the line satisfies certain Diophantine conditions.

Almost every point on a line with irrational slope is not Dirichlet-improvable.

The results give a dynamical criterion for Diophantine properties of points on lines.

Abstract

Let $X=\text{SL}_3(\mathbb{R})/\text{SL}_3(\mathbb{Z})$, and $g_t=\text{diag}(e^{2t}, e^{-t}, e^{-t})$. Let $\nu$ denote the push-forward of the normalized Lebesgue measure on a segment of a straight line in the expanding horosphere of $\{g_t\}_{t>0}$, under the map $h\mapsto h\text{SL}_3(\mathbb{Z})$ from $\text{SL}_3(\mathbb{R})$ to $X$. We give explicit necessary and sufficient Diophantine conditions on the line for equidistribution of each of the following families of measures on $X$: (1) $g_t$-translates of $\nu$ as $t\to\infty$. (2) averages of $g_t$-translates of $\nu$ over $t\in[0,T]$ as $T\to\infty$. (3) $g_{t_i}$-translates of $\nu$ for some $t_i\to\infty$. We apply this dynamical result to show that Lebesgue-almost every point on the planar line $y=ax+b$ is not Dirichlet-improvable if and only if $(a,b)\notin\mathbb{Q}^2$.