A note on exceptional sets in Erdös–Rényi limit theorem

For \(x\in [0,1]\), the run-length function \(r_n(x)\) is defined as the length of longest run of 1’s among the first n dyadic digits in the dyadic expansion of x. We study the Hausdorff dimension of the exceptional set in Erdös–Rényi limit theorem. Let \(\varphi :{{\mathbb {N}}}\rightarrow (0,\infty )\) be a monotonically increasing function with \(\lim _{n\rightarrow \infty }\varphi (n)=\infty \) and \(0\le \alpha \le \beta \le \infty \), define

$$\begin{aligned} E_{\alpha ,\beta }^\varphi =\Big \{x\in [0,1]:\, \liminf _{n\rightarrow \infty } \dfrac{r_n(x)}{\varphi (n)}=\alpha , \limsup _{n\rightarrow \infty } \frac{r_n(x)}{\varphi (n)}=\beta \Big \}. \end{aligned}$$

We prove that \(E_{\alpha ,\beta }^\varphi \) has Hausdorff dimension one if \(\lim _{n,p\rightarrow \infty }\frac{\varphi (n+p)-\varphi (n)}{p}=0\) and that \(E_{0,\infty }^\varphi \) is residual in [0,1] when \(\liminf _{n\rightarrow {\infty }}\frac{\varphi (n)}{n}=0\).