Abstract
Let A be a language with infinite number of words over the alphabet Σ={0,1} and let (xi)i=1+∞ be the digits sequence in the unique terminating dyadic expansion of x∈[0,1). The maximal run-length function ln(x,A) of x over A is defined byln(x,A)=max{k:xi+1xi+2⋯xi+k∈Afor some0≤i≤n−k}. In a general setting, Wu (2024) [18] proved a metrical property on the asymptotic behavior of ln(x,A), which extends the famous Erdős-Rényi limit theorem that represents for the special case A={1k:k≥1}. Under the assumption that A is factorial (i.e. the language A is closed under subword), Wu studied the exceptional setE={x∈[0,1):limn→∞ln(x,A)mn=0andlimn→∞ln(x,A)=∞}, and he obtained a lower bound of the Hausdorff dimension of E with some conditions on mn. In this paper, we determine the Hausdorff dimension of E without any additional assumption on mn and we obtain the Hausdorff dimension of two other kinds of exceptional sets of elements whose maximal run-length functions have different growth rates. Our results significantly extend the existing dimension results in this topic.
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