The growth speed for the product of consecutive digits in Luroth expansions

For x is an element of [0, 1), let [d(1)( x), d(2)(x),...] be its Luroth expansion and {p(n)(x)/qn (x), n >= 1} be the sequence of convergents of x. For alpha, beta is an element of [0,+infinity) with a <= beta, we define the exceptional sets E(beta) = {x is an element of [0, 1) : lim sup (n ->infinity) log (d(n)(x)d(n+1)(x))/log q(n)(x) = beta} and F(alpha, beta) = {x [0, 1) : lim inf(n ->infinity) log (d(n)(x)d(n+1)(x))/log qn(x) = a, lim sup(n ->infinity) log d(n)(x)d(n+1)(x))/log q(n)(x) = beta}. In this paper, we completely determine the Hausdorff dimension of sets E(beta) and F(alpha, beta).