LAUSR

Let [a 1(x), a 2(x), …, a n (x), …] be the continued fraction expansion of x ∈ [0, 1) and q n (x) be the denominator of the nth convergent. The study of relative growth rate of the product of partial quotients a n+1(x)a n (x) compared with q n (x) originated from the improvability of Dirichlet’s theorem. In this note, we prove that, for any 0 ⩽ α ⩽ β ⩽ +∞, the Hausdorff dimension of the following set Fα,β=x∈[0,1):lim infn→∞log(an+1(x)an(x))logqn(x)=α,lim supn→∞log(an+1(x)an(x))logqn(x)=β is 2β+2+β2+4 or 2β+2 according to α > 0 or α = 0, respectively. This result extends an earlier result of Huang and Wu as well as gives insights on the metric theory of Dirichlet non-improvable sets.