Equidistribution on homogeneous spaces and the distribution of approximates in Diophantine approximation
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Abstract:
The present paper is concerned with equidistribution results for certain flows on homogeneous spaces and related questions in Diophantine approximation. First, we answer in the affirmative, a question raised by Kleinbock, Shi, and Weiss regarding equidistribution of orbits of arbitrary lattices under diagonal flows and with respect to unbounded functions. We then consider the problem of Diophantine approximation with respect to rationals in a fixed number field. We prove a number field analogue of a famous result of W. M. Schmidt which counts the number of approximates to Diophantine inequalities for a certain class of approximating functions. Further we prove “spiraling” results for the distribution of approximates of Diophantine inequalities in number fields. This generalizes the work of Athreya, Ghosh, and Tseng as well as Kleinbock, Shi, and Weiss.References
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Additional Information
- Mahbub Alam
- Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Mumbai, 400005 India
- Email: mahbub@math.tifr.res.in
- Anish Ghosh
- Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Mumbai, 400005 India
- MR Author ID: 766213
- Email: ghosh@math.tifr.res.in
- Received by editor(s): February 18, 2019
- Received by editor(s) in revised form: August 23, 2019
- Published electronically: February 11, 2020
- Additional Notes: The second author was supported by a grant from the Indo-French Centre for the Promotion of Advanced Research; a Department of Science and Technology, Government of India Swarnajayanti fellowship and a MATRICS grant from the Science and Engineering Research Board.
- © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 3357-3374
- MSC (2010): Primary 37A17; Secondary 11K60
- DOI: https://doi.org/10.1090/tran/7997
- MathSciNet review: 4082241