THE GROWTH RATE OF THE DIGITS IN THE LUROTH EXPANSIONS

For any x is an element of (0, 1], let x = 1/d(1) + 1/d(1)(d(1) - 1)d(2) + ... + 1/d(1)(d(1) - 1) ... d(n-1) (d(n-1) - 1)d(n) + ... be its Luroth expansion with digits {d(j) >= 2,for all j >= 1}. This paper is concerned with the growth rate of the digits in the Luroth expansions. Let psi : N -> R+ be a function satisfying psi(n + 1) - psi(n) -> infinity as n -> infinity and lim(n ->infinity) (psi(n +1)/psi(n) = b >= 1. In this paper, we consider the set E(psi) := { x is an element of (0, 1] : lim(n ->infinity) 1/psi(n) Sigma(n)(j=1) log d(j) (x) = 1} , and we quantify the size of E(psi) in the sense of Hausdorff dimension. As applications, we get the Hausdorff dimensions of the sets of points for which Sigma(n)(j=1) log d(j) (x) grows with polynomial and exponential rate.