The Hausdorff dimension of level sets described by Erdös-Rényi average

Let 0≤α≤1 and ϕ be an integer function defined on N∖{0} satisfying 1≤ϕ(n)≤n. Define the level setERϕ(α)={x∈[0,1]:limn→∞⁡An,ϕ(n)(x)=α}, where An,ϕ(n)(x) is the (n,ϕ(n))-Erdös-Rényi average of x∈[0,1]. In this paper, we will give descriptions for the Hausdorff dimension of ERϕ(α) under the assumption ϕ(n)→∞ as n→∞, which complement simultaneously an early classic result of Besicovitch and the new strong law of large number established by P. Erdös and A. Rényi. Moreover, for the case ϕ(n)=M ultimately, where M≥1 is an integer, the Hausdorff dimension of ERϕ(α) is also determined by us in the last section.