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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.
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A dichotomy phenomenon for bad minus normed Dirichlet
Abstract 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.
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- Award ID(s):
- 2155111
- PAR ID:
- 10441557
- Publisher / Repository:
- Oxford University Press (OUP)
- Date Published:
- Journal Name:
- Mathematika
- Volume:
- 69
- Issue:
- 4
- ISSN:
- 0025-5793
- Format(s):
- Medium: X Size: p. 1145-1164
- Size(s):
- p. 1145-1164
- Sponsoring Org:
- National Science Foundation
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