Level sets of partial maximal digits for Luroth expansion

Let [d(1)(x), d(2)(x),..., d(n)(x),...] be the Luroth expansion of x. (0, 1). This paper is concerned with the growth rate of the partial maximum L-n(x) = max(1)<k<n{dk(x)}. We completely determined the Hausdorff dimension of the set E-phi ={x epsilon(0, 1) : lim (n infinity) L-n(x)/phi(n) = 1} when phi(n) tends to infinity with polynomial or exponential rates.