let $$X=\left\{ \sum_{n=1}^{\infty}a_{n}2^{-n}:a_{n}\in\left\{ 0,1\right\} ,\liminf\frac{1}{n}\sum_{i=1}^{n}a_{i}<\limsup\frac{1}{n}\sum_{i=1}^{n}a_{i}\right\} $$
I'm studying fractal geometry and I've encountered the following question: Is there an $\alpha$-regular measure giving X positive measure?
Now, I've noticed that the Lebesgue measure of X must be zero, simply by looking at it's complement, and by the LLN the limit $\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}a_{i}$ exists almost surely, hence $X^{c}$ is of Lebesgue measure 0. I'm not sure how to proceed, and would be happy for any suggestions. In general I'm trying to find the Hausdorff dimension of X.