Hausdorff dimension in inhomogeneous Diophantine approximation

2018/05/26 by Bugeaud, Yann, Kim, Dong Han, Lim, Seonhee +1
#37E10 #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT) #Primary 11K60 #Secondary 28A80

paper · doi:10.48550/arxiv.1805.10436

Abstract

Let α be an irrational real number. We show that the set of ε-badly approximable numbers Bad^ε (α) := \x∈ [0,1] : \liminf_{|q| → ∞} |q| ⋅ ‖ qα-x ‖ ≥ ε \ has full Hausdorff dimension for some positive ε if and only if α is singular on average. The condition is equivalent to the average (1)/(k) ∑_{i=1, ⋯, k} log a_i of the logarithms of the partial quotients a_i of α going to infinity with k. We also consider one-sided approximation, obtain a stronger result when a_i tends to infinity, and establish a partial result in higher dimensions.

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