Uniform Diophantine approximation related to b-ary and β-expansions

Original · EN

Let b≥ 2 be an integer and a real number. Among other results, we compute the Hausdorff dimension of the set of real numbers ξ with the property that, for every sufficiently large integer N, there exists an integer n such that 1 ≤ n ≤ N and the distance between bⁿ ξ and its nearest integer is at most equal to b- N. We further solve the same question when replacing bⁿξ by Tⁿβξ, where Tβ denotes the classical β-transformation.