Search for: All records

This paper is a sequel to [Monatsh. Math. 194 (2021) 523–554] in which results of that paper are generalized so that they hold in the setting of inhomogeneous Diophantine approximation. Given any integers [Formula: see text] and [Formula: see text], any [Formula: see text], and any homogeneous function [Formula: see text] that satisfies a certain nonsingularity assumption, we obtain a biconditional criterion on the approximating function [Formula: see text] for a generic element [Formula: see text] in the [Formula: see text]-orbit of [Formula: see text] to be (respectively, not to be) [Formula: see text]-approximable at [Formula: see text]: that is, for there to exist infinitely many (respectively, only finitely many) [Formula: see text] such that [Formula: see text] for each [Formula: see text]. In this setting, we also obtain a sufficient condition for uniform approximation. We also consider some examples of [Formula: see text] that do not satisfy our nonsingularity assumptions and prove similar results for these examples. Moreover, one can replace [Formula: see text] above by any closed subgroup of [Formula: see text] that satisfies certain integrability axioms (being of Siegel and Rogers type) introduced by the authors in the aforementioned previous paper. 

Given a norm ν on , the set of ν‐Dirichlet improvable numbers was defined and studied in the papers (Andersen and Duke,Acta Arith. 198 (2021) 37–75 and Kleinbock and Rao,Internat. Math. Res. Notices2022 (2022) 5617–5657). When ν is the supremum norm, , where is the set of badly approximable numbers. Each of the sets , like , is of measure zero and satisfies the winning property of Schmidt. Hence for every norm ν, is winning and thus has full Hausdorff dimension. In this article, we prove the following dichotomy phenomenon: either or else has full Hausdorff dimension. We give several examples for each of the two cases. The dichotomy is based on whether thecritical locusof ν intersects a precompact ‐orbit, where is the one‐parameter diagonal subgroup of acting on the spaceXof unimodular lattices in . Thus, the aforementioned dichotomy follows from the following dynamical statement: for a lattice , either is unbounded (and then any precompact ‐orbit must eventually avoid a neighborhood of Λ), or not, in which case the set of lattices inXwhose ‐trajectories are precompact and contain Λ in their closure has full Hausdorff dimension.

 

Abstract In the study of some dynamical systems the limsup set of a sequence of measurable sets is often of interest. The shrinking targets and recurrence are two of the most commonly studied problems that concern limsup sets. However, the zero–one laws for the shrinking targets and recurrence are usually treated separately and proved differently. In this paper, we introduce a generalized definition that can specialize into the shrinking targets and recurrence; our approach gives a unified proof of the zero–one laws for the two problems. 

Abstract Let G be a Lie group, let $\Gamma \subset G$ be a discrete subgroup, let $X=G/\Gamma $ and let f be an affine map from X to itself. We give conditions on a submanifold Z of X that guarantee that the set of points $x\in X$ with f -trajectories avoiding Z is hyperplane absolute winning (a property which implies full Hausdorff dimension and is stable under countable intersections). A similar result is proved for one-parameter actions on X . This has applications in constructing exceptional geodesics on locally symmetric spaces and in non-density of the set of values of certain functions at integer points.