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On numbers badly approximable by dyadic rationals

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Abstract

We consider a problem originating both from circle coverings and badly approximable numbers in the case of dyadic diophantine approximation. For the unit circle \( \mathbb{S} \) we give an elementary proof that the set {x ∈ \( \mathbb{S} \): 2n x ≥ c (mod 1) n ≥ 0} is a fractal set whose Hausdorff dimension depends continuously on c and is constant on intervals which form a set of Lebesgue measure 1. Hence it has a fractal graph. We completely characterize the intervals where the dimension remains unchanged. As a consequence we can describe the graph of c ↦ dim H {x ∈ [0; 1]: x − m/2n < c/2n (mod 1) finitely often}.

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  1. Centre for Mathematical Sciences, Lund University LTH, P.O.Box 118, SE-22100, Lund, Sweden

    Johan Nilsson

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  1. Johan Nilsson
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Nilsson, J. On numbers badly approximable by dyadic rationals. Isr. J. Math. 171, 93–110 (2009). https://doi.org/10.1007/s11856-009-0042-9

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  • DOI: https://doi.org/10.1007/s11856-009-0042-9

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