Let $E$ be a self-similar set with similarities ratio $r_j(0\le j\le m-1)$ and Hausdorff dimension $s$, let $\overrightarrow{p}(p_0,p_1)...p_{m-1}$ be a probability vector. The Besicovitch-type subset of $E$ is defined as$$E\left(\overrightarrow{p}\right)=\left\{x\in E:\underset{n\to \infty }{lim}\frac{1}{n}\sum _{k=1}^n{\chi }_j\left(x_k\right)=p_j,0\le j\le m-1\right\},$$where ${\chi }_j$ is the indicator function of the set $\left\{j\right\}$. Let $\alpha ={dim}_H\left(E\left(\overrightarrow{p}\right)\right)={dim}_P\left(E\left(\overrightarrow{p}\right)\right)=\frac{{\sum }_{j=0}^{m-1}{p}_{j}log{p}_j}{{\sum }_{j=0}^{m-1}{p}_{i}log{r}_j}$ and $g$ be a gauge function, then we prove in this paper:(i) If $\overrightarrow{p}=(r_0^s,r_1^s,\cdots ,r_{m-1}^s)$, then$${\mathscr{H}}^s\left(E\left(\overrightarrow{p}\right)\right)={\mathscr{H}}^s\left(E\right),{𝒫}^s\left(E\left(\overrightarrow{p}\right)\right)={𝒫}^s\left(E\right),$$moreover both of ${\mathscr{H}}^s\left(E\right)$ and ${𝒫}^s\left(E\right)$ are finite positive;(ii) If $\overrightarrow{p}$ is a positive probability vector other than $(r_0^s,r_1^s,\cdots ,r_{m-1}^s)$, then the gauge functions can be partitioned as follows$${\mathscr{H}}^g\left(E\left(\overrightarrow{p}\right)\right)=+\infty \iff \underset{t\to 0}{\overline{\lim }}\frac{logg\left(t\right)}{logt}\le \alpha ;{\mathscr{H}}^g\left(E\left(\overrightarrow{p}\right)\right)=0⟺\underset{t\to 0}{\overline{\lim }}\frac{logg\left(t\right)}{logt}\>\alpha ,$$$$...$$

We show that the set of numbers with bounded Lüroth expansions (or bounded Lüroth series) is winning and strong winning. From either winning property, it immediately follows that the set is dense, has full Hausdorff dimension, and satisfies a countable intersection property. Our result matches the well-known analogous result for bounded continued fraction expansions or, equivalently, badly approximable numbers. We note that Lüroth expansions have a countably infinite Markov partition,...

For a given sequence a boundedly expressible set is introduced. Three criteria concerning the Hausdorff dimension of such sets are proved.