北海道力学系webセミナー: Hausdorff dimension of sets with restricted, slowly growing partial quotients ( 高橋 博樹 （慶應理工）)

Time：2020年7月28日(火)　15:00～16:00

Speaker：高橋 博樹 （慶應理工）

Title：Hausdorff dimension of sets with restricted, slowly growing partial quotients

Abstract：I. J. Good (1941) showed that the set of irrational numbers in $(0,1)$ whose partial quotients $a_n$ tend to infinity is of Hausdorff dimension 1/2. A number of related results impose restrictions of the type $a_n \in B$ or $a_n > f(n)$, where $B$ is an infinite subset of $\mathbb N$ and $f$ is a rapidly growing function with $n$. We show that, for an arbitrary $B$ and an arbitrary $f$ with values in $[\min B,\infty)$ and tending to infinity, the set of irrational numbers in $(0,1)$ such that $a_n \in B$, $a_n < f(n)$ for every $n \in \mathbb N$ and $a_n\to\infty$ as $n \to \infty$ is of Hausdorff dimension $c(B)/2$, where $c(B)$ is the exponent of convergence of $B$.

世話役: 豊川（北大理）, 行木（北大理）, 中村（北見工大）

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