Extremely Exceptional Sets on Run-Length Function for Reals in Beta-Dynamical System

The extremely exceptional set for the run-length function in the beta-dynamical system is investigated in this study. For any real x in $(0,1]$, the run-length function related to x that measures the maximal length of the initial digit sequence of the $\beta$-expansion of x appears consecutively among the first n digits of the $\beta$-expansion of another real number y in $(0,1]$. The extremely exceptional set consists of all real numbers y with run-length exhibiting extreme oscillatory behavior: the limit inferior of the ratio of the run-length function to the logarithm base $\beta$ of n is zero, while the limit superior of this same ratio is infinity. We prove that the Hausdorff dimension of this set is either 0 or 1, determined solely by the asymptotic scaling of the basic intervals containing x. Crucially, for all x belonging to $(0,1]$, the set is residual in $[0,1]$, which implies that its boxing dimension is 1, which generalizes some known results.