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.

设 [ a ₁ ( x ), a ₂ ( x ), …, a _n ( x ), …] 是x ∈ [0, 1) 的连分数展开式，q _n ( x ) 是第n个的分母收敛。偏商a _{n +1} ( x ) a _n ( x )与q _n ( x )的乘积的相对增长率研究 ) 源于狄利克雷定理的可改进性。在本笔记中，我们证明，对于任何 0 ⩽ α ⩽ β ⩽ +∞，以下集合的 Hausdorff 维数分别是或根据α > 0 或α = 0。该结果扩展了 Huang 和 Wu 的早期结果，并提供了对 Dirichlet 不可改进集的度量理论的见解。