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Mosc. Math. J., 2005, Volume 5, Number 4, Pages 747–766 (Mi mmj220)  

This article is cited in 40 scientific papers (total in 40 papers)

On exponents of homogeneous and inhomogeneous Diophantine approximation

Ya. Bugeauda, M. Laurentb

a University Louis Pasteur
b Institut de Mathématiques de Luminy

Abstract: In Diophantine Approximation, inhomogeneous problems are linked with homogeneous ones by means of the so-called Transference Theorems. We revisit this classical topic by introducing new exponents of Diophantine approximation. We prove that the inhomogeneous exponent of approximation to a generic point in $\mathbb R^n$ by a system of $n$ linear forms is equal to the inverse of the uniform homogeneous exponent associated to the system of dual linear forms.

Key words and phrases: Diophantine approximation, measures of homogeneous and inhomogeneous approximation, uniform exponents, spectra.

DOI: https://doi.org/10.17323/1609-4514-2005-5-4-747-766

Full text: http://www.ams.org/.../abst5-4-2005.html
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MSC: 11J20, 11J13, 11J82
Received: November 4, 2004
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Citation: Ya. Bugeaud, M. Laurent, “On exponents of homogeneous and inhomogeneous Diophantine approximation”, Mosc. Math. J., 5:4 (2005), 747–766

Citation in format AMSBIB
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\by Ya.~Bugeaud, M.~Laurent
\paper On exponents of homogeneous and inhomogeneous Diophantine approximation
\jour Mosc. Math.~J.
\yr 2005
\vol 5
\issue 4
\pages 747--766
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\crossref{https://doi.org/10.17323/1609-4514-2005-5-4-747-766}
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Bugeaud Ya., Laurent M., “Exponents of diophantine approximation”, Diophantine Geometry, Proceedings, CRM Series, 4, 2007, 101–121  mathscinet  zmath  isi
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    5. Bugeaud Ya., Evertse J.-H., “Approximation of complex algebraic numbers by algebraic numbers of bounded degree”, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 8:2 (2009), 333–368  mathscinet  zmath  isi
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    17. Schmidt W.M., Summerer L., “Diophantine Approximation and Parametric Geometry of Numbers”, Mon.heft. Math., 169:1 (2013), 51–104  crossref  mathscinet  zmath  isi  scopus
    18. Bounemoura A., Fischler S., “A Diophantine Duality Applied to the Kam and Nekhoroshev Theorems”, Math. Z., 275:3-4 (2013), 1135–1167  crossref  mathscinet  zmath  isi  scopus
    19. Ghosh A., Gorodnik A., Nevo A., “Diophantine Approximation and Automorphic Spectrum”, Int. Math. Res. Notices, 2013, no. 21, 5002–5058  crossref  mathscinet  zmath  isi  scopus
    20. Shapira U., “Grids with Dense Values”, Comment. Math. Helv., 88:2 (2013), 485–506  crossref  mathscinet  zmath  isi  scopus
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    22. Fischler S., Hussain M., Kristensen S., Levesley J., “a Converse To Linear Independence Criteria, Valid Almost Everywhere”, Ramanujan J., 38:3 (2015), 513–528  crossref  mathscinet  zmath  isi  scopus
    23. Dani S.G., Laurent M., Nogueira A., “Multi-Dimensional Metric Approximation By Primitive Points”, Math. Z., 279:3-4 (2015), 1081–1101  crossref  mathscinet  zmath  isi  scopus
    24. Wu Y., Shamai (Shitz) Shlomo, Verdu S., “Information Dimension and the Degrees of Freedom of the Interference Channel”, IEEE Trans. Inf. Theory, 61:1 (2015), 256–279  crossref  mathscinet  zmath  isi  scopus
    25. Ghosh A., Gorodnik A., Nevo A., “Diophantine Approximation Exponents on Homogeneous Varieties”, Recent Trends in Ergodic Theory and Dynamical Systems, Contemporary Mathematics, 631, eds. Bhattacharya S., Das T., Ghosh A., Shah R., Amer Mathematical Soc, 2015, 181–200  crossref  mathscinet  zmath  isi
    26. Stotz D., Bolcskei H., “Degrees of Freedom in Vector Interference Channels”, IEEE Trans. Inf. Theory, 62:7 (2016), 4172–4197  crossref  mathscinet  zmath  isi  scopus
    27. Harrap S., Moshchevitin N., “a Note on Weighted Badly Approximable Linear Forms”, Glasg. Math. J., 59:2 (2017), 349–357  crossref  mathscinet  zmath  isi  scopus
    28. Singhal L., “Diophantine Exponents For Standard Linear Actions of Sl2 Over Discrete Rings in C”, Acta Arith., 177:1 (2017), 53–73  crossref  mathscinet  zmath  isi  scopus
    29. Bank E., Nesharim E., Pedersen S.H., “Solution of Cassels' Problem on a Diophantine Constant Over Function Fields”, Int. Math. Res. Notices, 2017, no. 18, 5451–5474  crossref  isi  scopus
    30. Bengoechea P., Moshchevitin N., “Badly Approximable Points in Twisted Diophantine Approximation and Hausdorff Dimension”, Acta Arith., 177:4 (2017), 301–314  crossref  zmath  isi  scopus
    31. Ghosh A., Gorodnik A., Nevo A., “Best Possible Rates of Distribution of Dense Lattice Orbits in Homogeneous Spaces”, J. Reine Angew. Math., 745 (2018), 155–188  crossref  mathscinet  zmath  isi  scopus
    32. Fukshansky L., Moshchevitin N., “On An Effective Variation of Kronecker'S Approximation Theorem Avoiding Algebraic Sets”, Proc. Amer. Math. Soc., 146:10 (2018), 4151–4163  crossref  mathscinet  zmath  isi  scopus
    33. Marnat A., “About Jarnik'S-Type Relation in Higher Dimension”, Ann. Inst. Fourier, 68:1 (2018), 131–150  crossref  mathscinet  zmath  isi
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    35. Johannes Schleischitz, “Diophantine approximation in prescribed degree”, Mosc. Math. J., 18:3 (2018), 491–516  mathnet  crossref
    36. Kim D.H., Liao L., “Dirichlet Uniformly Well-Approximated Numbers”, Int. Math. Res. Notices, 2019:24 (2019), 7691–7732  crossref  mathscinet  isi
    37. Chow S., Technau N., “Higher-Rank Bohr Sets and Multiplicative Diophantine Approximation”, Compos. Math., 155:11 (2019), 2214–2233  crossref  mathscinet  zmath  isi
    38. Bugeaud Ya., Zhang Zh., “on Homogeneous and Inhomogeneous Diophantine Approximation Over the Fields of Formal Power Series”, Pac. J. Math., 302:2 (2019), 453–480  crossref  mathscinet  zmath  isi  scopus
    39. Bugeaud Ya., Cheung Y., Chevallier N., “Hausdorff Dimension and Uniform Exponents in Dimension Two”, Math. Proc. Camb. Philos. Soc., 167:2 (2019), 249–284  crossref  mathscinet  zmath  isi  scopus
    40. Ghosh A., Marnat A., “on Diophantine Transference Principles”, Math. Proc. Camb. Philos. Soc., 166:3 (2019), 415–431  crossref  mathscinet  zmath  isi  scopus
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