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Singular vectors on manifolds and fractals

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Abstract

We generalize Khintchine’s method of constructing totally irrational singular vectors and linear forms. The main result of the paper shows existence of totally irrational vectors and linear forms with large uniform Diophantine exponents on certain subsets of ℝn, in particular on any analytic submanifold of ℝn of dimension ≥2 which is not contained in a proper rational affine subspace.

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Acknowledgements

The authors were supported by NSF grant DMS-1600814, RFBR grant No. 18-01-00886 and BSF grant 2016256, respectively. This work was started during the second-named author’s visit to Brandeis University, whose hospitality is gratefully acknowledged. The authors are also grateful to the MATRIX institute for providing a stimulating atmosphere for a productive collaboration, and to the anonymous referee for taking a close look at the paper and catching some errors.

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Authors and Affiliations

  1. Department of Mathematics, Brandeis University, Waltham, MA, 02454-9110, USA

    Dmitry Kleinbock

  2. Moscow Center of Fundamental and Applied Mathematics, Leninskie Gory 1, Moscow, 1119991, Russia

    Nikolay Moshchevitin

  3. Astrakhan State University, Tatishcheva 20a, Astrakhan, 414056, Russia

    Nikolay Moshchevitin

  4. Department of Mathematics, Tel Aviv University, Ramat Aviv, Tel Aviv, 69978, Israel

    Barak Weiss

Authors
  1. Dmitry Kleinbock
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  2. Nikolay Moshchevitin
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  3. Barak Weiss
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Correspondence to Barak Weiss.

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Kleinbock, D., Moshchevitin, N. & Weiss, B. Singular vectors on manifolds and fractals. Isr. J. Math. 245, 589–613 (2021). https://doi.org/10.1007/s11856-021-2220-3

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  • DOI: https://doi.org/10.1007/s11856-021-2220-3

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