Equidistribution in the space of 3-lattices and Dirichlet-improvable vectors on planar 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 $(a,b)\notin {\mathbb{Q}}^2$.

arXiv comment35 pages, 2 figures