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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: fadiceam@gmail.com)
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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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