The Hausdorff dimension of level sets described by Erdos-Renyi average

Let 0 <= alpha <= 1 and phi. be an integer function defined on N \ {0} satisfying 1 <= phi(n) <= n. Define the level set ER phi(alpha) = {x is an element of [0, 1]: lim(n ->infinity) A(n,phi(n)) (x) = alpha}, where A(n,phi(n)) (x) is the (n, phi(n))-Erdos Renyi average of x is an element of [0,1]. In this paper, we will give descriptions for the Hausdorff dimension of ER phi(alpha) under the assumption phi(n) -> infinity as n -> infinity, which complement simultaneously an early classic result of Besicovitch and the new strong law of large number established by P. Eras and A. Renyi. Moreover, for the case phi(n) = M ultimately, where M >= 1 is an integer, the Hausdorff dimension of ER phi(alpha) is also determined by us in the last section. (C) 2017 Elsevier Inc. All rights reserved.