LAUSR

Abstract Let m ≥ 2 and 1 ≤ p ≤ m − 1 be two integers. Denote by Σ ¯ p ⊂ { 0 , 1 , … , m − 1 } the alphabet with p elements. In this paper we will introduce the run-length function R n p ( x ) of x ∈ [ 0 , 1 ] in Σ ¯ p and give a generalization to the classic Erdos–Renyi limit theorem. More precisely, we will prove that lim n → ∞ ⁡ R n p ( x ) / ( log m / p ⁡ n ) = 1 for almost all x ∈ [ 0 , 1 ] . Moreover, for the level set E m ( ϕ ) = { x ∈ [ 0 , 1 ] : lim n → ∞ ⁡ R n p ( x ) ϕ ( n ) = 1 } where ϕ is a positive function defined on N , its Hausdorff dimension is determined when the function ϕ is of some particular growth rates. As a result, for any 0 ≤ α ≤ ∞ , the level set in which the limit lim n → ∞ ⁡ R n p ( x ) / ( log m / p ⁡ n ) equals to α is of full Hausdorff dimension. It complements the generalized Erdos–Renyi limit theorem established by us.