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A NOTE ON THE HAUSDORFF DIMENSION OF SOME LIMINF SETS APPEARING IN SIMULTANEOUS DIOPHANTINE APPROXIMATION

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  05 December 2012

Faustin Adiceam
Faustin Adiceam*
Affiliation:
Department of Mathematics and Statistics, National University of Ireland at Maynooth, Maynooth, Co Kildare, Ireland (email: [email protected])
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Abstract

Let Q be an infinite set of positive integers. Denote by Wτ,n(Q) (respectively, Wτ,n) the set of points in dimension n≥1 that are simultaneously τ-approximable by infinitely many rationals with denominators in Q (respectively, in ℕ*). When n≥2 and τ>1+1/(n−1) , a non-trivial lower bound for the Hausdorff dimension of the liminf set Wτ,n ∖Wτ,n (Q) is established in the case where the set Q satisfies some divisibility properties. The computation of the actual value of this Hausdorff dimension and the one-dimensional analogue of the problem are also discussed.

MSC classification

Secondary: 11J83: Metric theory 11K60: Diophantine approximation
Type
Research Article
Information
Mathematika , Volume 59 , Issue 1 , January 2013 , pp. 56 - 64
Copyright
Copyright © University College London 2012

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References

[1]Babai, L. and Štefankovič, D., Simultaneous diophantine approximation with excluded primes. In Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms, Society for Industrial and Applied Mathematics (Philadelphia, 2004), 1123–1129.Google Scholar
[2]Bernik, V. I. and Dodson, M. M., Metric Diophantine Approximation on Manifolds, Cambridge University Press (Cambridge, 1999).CrossRefGoogle Scholar
[3]Borosh, I. and Fraenkel, A. S., A generalization of Jarnik’s theorem on Diophantine approximations. Nederl. Akad. Wet., Proc., Ser. A 75 (1972), 193–201.Google Scholar
[4]Bugeaud, Y., Approximation by Algebraic Numbers (Cambridge Tracts in Mathematics 160), Cambridge University Press (Cambridge, 2007).Google Scholar
[5]Dickinson, D. and Velani, S. L., Hausdorff measure and linear forms. J. Reine Angew. Math. 490 (1997), 1–36.Google Scholar
[6]Erdős, P., On the difference of consecutive terms of sequences defined by divisibility properties. Acta Arith. 12 (1966), 175–182.CrossRefGoogle Scholar
[7]Erdős, P. and Mahler, K., Some arithmetical properties of the convergents of a continued fraction. J. Lond. Math. Soc. 14 (1939), 12–18.CrossRefGoogle Scholar
[8]Harman, G., Metric Number Theory (London Mathematical Society Monographs, New Series 18) 1998.CrossRefGoogle Scholar
[9]Jarník, V., Über die simultanen diophantischen Approximationen. Math. Z. 33 (1931), 505–543.CrossRefGoogle Scholar
[10]Rynne, B. P., Hausdorff dimension and generalized simultaneous diophantine approximation. Bull. Lond. Math. Soc. 30(4) (1998), 365–376.CrossRefGoogle Scholar
[11]Rynne, B. P. and Dickinson, H., Hausdorff dimension and a generalized form of simultaneous diophantine approximation. Acta Arith. 93(1) (2000), 21–36.CrossRefGoogle Scholar