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Diophantine exponents, best approximation and badly approximable numbers

By Nikolay Moshchevitin

Appears in collection : Jean-Morlet Chair 2020 - Conference: Diophantine Problems, Determinism and Randomness / Chaire Jean-Morlet 2020 - Conférence : Problèmes diophantiens, déterminisme et aléatoire

We will discuss recent progress in analysis of uniform and ordinary Diophantine exponents $\hat\omega $ and $\omega$ for linear Diophantine approximation as well as some applications of the related methods. In particular, we give a new criterion for badly approximable vectors in $\mathbb{R}^{d}$ the behavior of the best approximation vectors in the sense of simultaneous approximation and in the sense of linear form. It turned out that compared to the one-dimensional case our criterion is rather unusual. We apply this criterion to the analysis of Dirichlet spectrum for simultaneous Diophantine approximation.

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Citation data

  • DOI 10.24350/CIRM.V.19688203
  • Cite this video Moshchevitin, Nikolay (23/11/2020). Diophantine exponents, best approximation and badly approximable numbers. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19688203
  • URL https://dx.doi.org/10.24350/CIRM.V.19688203

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Bibliography

  • MARNAT, Antoine et MOSHCHEVITIN, Nikolay G. An optimal bound for the ratio between ordinary and uniform exponents of Diophantine approximation. Mathematika, 2020, vol. 66, no 3, p. 818-854. - https://doi.org/10.1112/mtk.12045
  • AKHUNZHANOV, Renat et MOSHCHEVITIN, Nikolay. On badly approximable numbers. arXiv preprint arXiv:2002.00433, 2020. - https://arxiv.org/abs/2002.00433

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