Metrical Properties of Functions of Consecutive Digits in Lüroth Series

Recently, the growth of the products of L¨ uroth quotients di(x) in the L¨ uroth expansion of a real number x was studied in connection with improvements to Dirichlet’s theorem. In this paper, for a non-decreasing positive measurable function F(x1,...,xm) and a function ϕ : N → R>0, we consider the following set: EF(ϕ) = {x ∈ [0,1] : F(dn(x),...,dn+m−1(x)) ≥ ϕ(n) for infinitely many n ∈ N}, and obtain its Lebesgue measure λ(EF(ϕ)). As an application of our result, we consider the case when F(x1,...,xm) = x1 +··· +xm.