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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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