Limit theorems for sums of products of consecutive partial quotients of continued fractions

Abstract

Abstract Let [a 1(x), a 2(x), …, a n (x), …] be the continued fraction expansion of an irrational number x ∈ (0, 1). The study of the growth rate of the product of consecutive partial quotients a n (x)a n+1(x) is associated with the improvements to Dirichlet’s theorem (1842). We establish both the weak and strong laws of large numbers for the partial sums S n ( x ) = ∑ i = 1 n a i ( x ) a i + 1 ( x ) as well as, from a multifractal analysis point of view, investigate its increasing rate. Specifically, we prove the following results: • For any ϵ > 0, the Lebesgue measure of the set x ∈ ( 0 , 1 ) : S n ( x ) n log 2 n − 1 2 log 2 ⩾ ϵ tends to zero as n to infinity. • For Lebesgue almost all x ∈ (0, 1), lim n → ∞ S n ( x ) − max 1 ⩽ i ⩽ n a i ( x ) a i + 1 ( x ) n log 2 n = 1 2 log 2 . • The Hausdorff dimension of the set E ( ϕ ) ≔ x ∈ ( 0 , 1 ) : lim n → ∞ S n ( x ) ϕ ( n ) = 1 is determined for a range of increasing functions ϕ : N → R + .