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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 setis determined for any 
\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*}
$ \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.
