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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))

2020年728日 開催

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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