Metrical properties of products formed by consecutive partial quotients in Luroth expansions

For x is an element of (0, 1], let [d(1)(x),..., d(x)(2),...] be its Luroth expansion, and {(2)P-k(x)/q(k)(x)}(k >= 1) be its convergents. Brown-Sarre et al. considered the metrical properties of products derived from sequential partial quotients raised to different powers. More precisely, for any t=(t(o),..., t(m)) is an element of R-+(m+1), they examined the metrical characteristics of the set Epsilon(iota)(psi) = {x is an element of (0,1]: d(k)(o)(t)(x)d(k+1)(1)(t) (x) ... d(k+m)(m)(t) (x)>=psi(k) for i.mk}, where m represents a positive integer, psi: N ->(1,infinity) is a function. In this paper, we will proceed with the study of these issues. Namely, the primary aim of this research is to explore the metrical properties of the following sets D-t (psi) = {x is an element of (0,1): c(k), k >= 1} and Ft (psi) = {x is an element of (0,1]: d(k)(o)(t)(x)d(k+1)(1)(t) (x) ... d(k+m)(m)(t) (x)>=psi (q(k)(x)) for i.m. k}. Moreover, we also give due consideration to the Hausdorff dimension of the exceptional set associated with the convergence exponent C-m (x) = inf {1 >= 0 : Sigma(infinity)(k=1) (d(k)(x)d(k+1)(x)... d(k+m) (x)(-l) <infinity}.