The Fine Structure of Dyadically Badly Approximable Numbers

الأصل · EN

We consider badly approximable numbers in the case of dyadic diophantine approximation. For the unit circle S and the smallest distance to an integer · we give elementary proofs that the set F(c) = {x ∈ S: 2ⁿx ≥ c, n≥ 0} is a fractal set whose Hausdorff dimension depends continuously on c, is constant on intervals which form a set of Lebesgue measure 1 and is self-similar. Hence it has a fractal graph. Moreover, the dimension of F(c) is zero if and only if c≥ 1-2τ, where τ is the Thue-Morse constant. We completely characterise the intervals where the dimension remains unchanged. As a consequence we can completely describe the graph of c ₕ {x∈[0,1]: x-m/2ⁿ< c/2ⁿ finitely often}.