Best possible rates of distribution of dense lattice orbits in homogeneous spaces

Abstract

<jats:title>Abstract</jats:title> <jats:p>This paper establishes upper and lower bounds on the speed of approximation in a wide range of natural Diophantine approximation problems. The upper and lower bounds coincide in many cases, giving rise to optimal results in Diophantine approximation which were inaccessible previously. Our approach proceeds by establishing, more generally, upper and lower bounds for the rate of distribution of dense orbits of a lattice subgroup Γ in a connected Lie (or algebraic) group <jats:italic>G</jats:italic>, acting on suitable homogeneous spaces <jats:italic>G</jats:italic>/<jats:italic>H</jats:italic>. The upper bound is derived using a quantitative duality principle for homogeneous spaces, reducing it to a rate of convergence in the mean ergodic theorem for a family of averaging operators supported on <jats:italic>H</jats:italic> and acting on <jats:italic>G</jats:italic>/Γ. In particular, the quality of the upper bound on the rate of distribution we obtain is determined explicitly by the spectrum of <jats:italic>H</jats:italic> in the automorphic representation on <jats:inline-formula id="j_crelle-2016-0001_ineq_9999"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_crelle-2016-0001_eq_0123.png" /> <jats:tex-math>L^{2}</jats:tex-math> </jats:alternatives> </jats:inline-formula>(Γ <jats:inline-formula id="j_crelle-2016-0001_ineq_9998"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_crelle-2016-0001_eq_0133.png" /> <jats:tex-math>\setminus</jats:tex-math> </jats:alternatives> </jats:inline-formula> <jats:italic>G</jats:italic>). We show that the rate is best possible when the representation in question is tempered, and show that the latter condition holds in a wide range of examples.</jats:p>

Shrinking targets for semisimple groups

“…We will call such a triple a tempered triple . An extensive list of tempered triples is provided in , and for the readers convenience we recall some of their examples below. (1)Consider and such that the representation of in is non‐trivial and irreducible.…”

Approximation of points in the plane by generic lattice orbits

“…, which is slightly better. However, our result holds for any lattice, and moreover, the method of proof generalizes to deal with the general problem of lattice action on homogenous spaces, thus answering the question of uniformity on a co-null set of orbits raised in [GGN14].…”

Diophantine exponents for standard linear actions of ${\rm SL}_2$ over discrete rings in $\mathbb {C}$

“…A work in the same spirit for SL 2 (Z) action on R 2 \ {0} was carried out by Maucourant and Weiss [10] with much weaker estimates than those of Laurent and Nogueira [8]. Applicable in a broader framework, the machinery of Ghosh, Gorodnik, and Nevo [5,6] is vastly superior and gives the values of exponents for an array of lattice actions on homogeneous varieties of connected almost simple, semisimple algebraic groups (see, in particular [5,6]). However, for them too, the lower bound for the uniform exponentμ Γ as in Def.…”

“…In recent times, mathematicians have been interested in understanding more generally the nature of dense orbits for the action of a group G on a homogeneous space X. In this respect, see the works of Ghosh, Gorodnik, and Nevo [5,6] where they relate the rate of approximation by 'rational points' on a homogeneous space X of a semisimple group G to the automorphic representations of G and compute the exact exponents for a number of examples. We point out upfront that their exponent κ Γ is exactly the inverse of the valuê µ Γ introduced in Def.…”