Abstract
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 is or 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.
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