Equidistribution of expanding translates of curves and Diophantine approximation on matrices

Abstract

We study the general problem of equidistribution of expanding translates of an analytic curve by an algebraic diagonal flow on the homogeneous space G/Γ of a semisimple algebraic group G. We define two families of algebraic subvarieties of the associated partial flag variety G/P , which give the obstructions to non-divergence and equidistribution. We apply this to prove that for Lebesgue almost every point on an analytic curve in the space of m × n real matrices whose image is not contained in any subvariety coming from these two families, the Dirichlet's theorem on simultaneous Diophantine approximation cannot be improved.The proof combines geometric invariant theory, Ratner's theorem on measure rigidity for unipotent flows, and linearization technique.

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

“…To analyze limiting distributions of sequences of translates of measures on homogeneous spaces a technique has been developed, where one applies Dani-Margulis and Kleinbock-Margulis non-divergence criteria, Ratner's theorem, and linearization method and reduces the problem to dynamics of subgroup actions on finite dimensional representations of semisimple groups, see [Sha09a,SY20]. In [Yan20], this kind of linear dynamics was analysed in a very general situation using invariant theory results due to Kempf [Kem78].…”

A measure estimate in geometry of numbers and improvements to Dirichlet's theorem

“…(This is so since in this case, the relevant "shrinking" target is in fact a fixed set of positive measure.) Shah's results have been generalized and strengthened in various directions [33,34,37,17]. In a recent breakthrough of Khalil and Luethi [14], the authors refined (1.14) (for the case when n = 1) by replacing Leb with a certain fractal measure, from which they deduced a complete analogue of Khintchine's theorem with respect to this fractal measure.…”

Groupes arithmétiques et approximation diophantienne

“…Nous verrons au chapitre suivant que la mesure de Lebesgue sur une sous-variété analytique est localement régulière. Admettant ce point pour l'instant, on retrouve comme cas particulier du théorème ci-dessus le résultat suivant, essentiellement dû à Pengyu Yang [46]…”

“…Ensuite, si V est une représentation rationnelle, et χ le plus haut poids apparaissant dans V , le premier minimum de a t sV (Z) est donné par λ 1 (a t sV (Z)) ≍ e χ(c(ats)) = e o(t) . la démonstration du théorème 7.3.1, nous aurons besoin de la proposition suivante, très voisine d'un résultat plus général de Pengyu Yang[46, Theorem 1.2]. Toutefois, la démonstration dans notre cas particulier est sensiblement plus simple, car on ne s'intéresse qu'aux vecteurs dans l'orbite d'un vecteur de plus haut poids ; en particulier nous n'aurons pas besoin des résultats de théorie géométrique des invariants dûs à Mumford[33] ou Kempf[17].Proposition 7.3.3 (Stabilité linéaire).…”