METRICAL PROPERTIES OF WEIGHTED PRODUCTS OF CONSECUTIVE LUROTH DIGITS

The Luroth expansion of a real number x is an element of(0,1] is the series x=1/d1+1/d1(d1-1)d2+1/d1(d1-1)d2(d2-1)d3+<middle dot><middle dot><middle dot>, with d(j)is an element of N >= 2 for all j is an element of N. Given m is an element of N, t= (t0,...,tm-1)is an element of Rm-1>0andany function Psi :N ->(1,infinity), define epsilon(t)(Psi) : =nx is an element of(0,1] :dt0n<middle dot><middle dot><middle dot>d(n+m)(tm-1)>=Psi(n) for infinitely many n is an element of N} We establish a Lebesgue measure dichotomy statement (a zero-one law) for epsilon(t)(Psi) under a natural non-removable condition lim in f(n)->infinity Psi(n)>1. Let B be given by log B: = lim in f(n)->infinity log(Psi(n))/n For any m is an element of N, we compute the Hausdorff dimension of epsilon(t)(Psi) when either B= 1 or B=infinity. We also compute the Hausdorff dimension of epsilon(t)(Psi) when1< B <infinity form= 2