Nonlinearity ( IF 1.7 ) Pub Date : 2021-03-09 , DOI: 10.1088/1361-6544/abe097 Jing Feng , Jian Xu
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.
中文翻译:
连分数论中具有一定顺序的狄利克雷不可改进数集该工作得到了国家自然科学基金11571127、11871208的支持。
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。