Exceptional sets related to the product of consecutive digits in Luroth expansions

Every real number x <euro> (0, 1] admits a L & uuml;roth expansion [d(1)(z), d(2)(x), ...] with da(1) N >= 2 being its digits. Let {,n> 1} be the sequence of con- vergents of the L & uuml;roth expansion of z. We study the growth rate of the product of consecutive digits relative to the denominator of the convergent for the L & uuml;roth expan- sion of an irrational number. More precisely, given a natural number m, we prove that the set<br /> E-m(beta) = {x <euro> (0, 1]: lim sup(n -> infinity)log (d(a) (x) d(n+1 (x) ... d)n+m(x)/log q(n (x) = beta})<br /> and the set<br /> log (2)<br /> E-m (beta) = {z <euro> (0,1]: lim sup(n -> infinity log) (d(a)(x) d(n+1 1(x) .. dn+m (x)) /log q)n((x) >= beta}) share the same Hausdorff dimension for 8 >= 0. It significantly generalises the existing results on the Hausdorff dimension of E-1(beta) and E-1(beta).