Hausdorff dimension of sets defined by almost convergent binary expansion sequences

Abstract

Abstract In this paper, we study the Hausdorff dimension of sets defined by almost convergent binary expansion sequences. More precisely, the Hausdorff dimension of the following set \begin{align*} \bigg\{x\in[0,1)\;:\;\frac{1}{n}\sum_{k=a}^{a+n-1}x_{k}\longrightarrow\alpha\textrm{ uniformly in }a\in\mathbb{N}\textrm{ as }n\rightarrow\infty\bigg\} \end{align*} is determined for any $ \alpha\in[0,1] $ . This completes a question considered by Usachev [Glasg. Math. J. 64 (2022), 691–697] where only the dimension for rational $ \alpha $ is given.