A note on the largest digits in Luroth expansion

It is well known that every x epsilon (0, 1] can be expanded into an infinite Luroth series with the form of x = 1/d(1)(x) + center dot center dot center dot + 1/(d(1)(x)-1)(d(1)(x)) center dot center dot center dot (d(n-1)(x)-1) d(n-1)(x)dn(x) + center dot center dot center dot, where d(n)(x) >= 2 and is called the nth digits of x for each n >= 1. In [Representations of Real Numbers by Infinite Series, Lecture Notes in Mathematics, Vol. 502 (Springer, New York, 1976)], Galambos showed that for Lebesgue almost all x epsilon (0, 1], lim(n ->+infinity) log Ln(x)/log n = 1, where L-n(x) = max{d(1)(x),...,d(n)(x)} denotes the largest digit among the first n ones of x. In this paper, we consider the Hausdorff dimension of the set E(alpha) - {x epsilon (0, 1] : lim(n ->infinity) log Ln(x)/log n = alpha} for any alpha >= 0.