We consider an analogue of Nakada's $α$-continued fraction transformation in the setting of continued fractions with odd partial quotients. More precisely, given $α\in [\frac{1}{2}(\sqrt{5}-1),\frac{1}{2}(\sqrt{5}+1)]$, we show that every irrational number $x\in I_α=[α-2,α)$ can be uniquely represented as $$ x= \cfrac{e_1 (x;α)}{d_1 (x;α) +\cfrac{e_2(x;α)}{d_2(x;α)+\cdots}} , $$ with $e_i(x;α) \in \{ \pm 1\}$ and $d_i(x;α) \in 2{\mathbb N} -1$ determined by the iterates of the transformation $$φ_α(x) := \frac{1}{| x|} - 2 \bigg[ \frac{1}{2| x|} +\frac{1-α}{2} \bigg]-1$$ of $I_α$. We also describe the natural extension of $φ_α$ and prove that the endomorphism $φ_α$ is exact.
Hausdorff dimension of sets with restricted, slowly growing partial quotients
I. J.Good（1941）证明了$(0,1)$中的无理数集合如果其偏导数$a_n$趋于无穷大，则它具有豪斯多夫维数$1/2$。一些相关的结果施加限制的类型$a_n\in B$或$a_n\geq f(n)$，其中$B$是一个无限子集的$\mathbb N$和$f$是一个快速增长的功能与$n$。我们证明了，对于任意的$B$和任意的$f$，其值在$[\min B,\infty)$中并趋于无穷大，$(0,1)$中的无理数集合使得\[ a_n\in B,\ a_n\leq f(n)\text{ for all $n\in\mathbb N$, and }a_n\to\infty\text{ as }n\to\infty\]具有豪斯多夫维数$τ(B)/2,$，其中$τ(B)$是$B$的收敛指数.
Extreme value theorem for geodesic flow on the quotient of the theta group
HUN-REN Alfréd Rényi Institute of Mathematics（HUN-REN 阿尔弗雷德·雷尼数学研究所）；Department of Mathematical Sciences, Seoul National University（首尔大学数学科学系）；Research Institute of Mathematics, Seoul National University（首尔大学数学研究所）
Limit theorems for sums of products of consecutive partial quotients of continued fractions
设$[a_1(x),a_2(x),\ldots, a_n(x), \ldots]$是无理数$x\in (0, 1)$的连分数展开式。对连续偏导数乘积$a_n(x)a_{n+1}(x)$的增长率的研究与狄利克雷定理（1842）的改进有关。我们建立了部分和$S_n(x)= \sum_{i=1}^n a_i(x)a_{i+1}(x)$的弱大数定律和强大数定律，并从多重分形分析的角度研究了其增长率。具体来说，我们证明了以下结果：\medskip \begin{itemize} \item For any $ε>0$, the Lebesgue measure of the set $$\left\{x\in(0, 1): \left|\frac{ S_n(x)}{n\log^2 n}-\frac1{2\log2}\right|\geq ε\right\}$$tends to zero as $n$ to infinity. \item For Lebesgue almost all $x\in (0,1)$, $$\lim\limits_{n\rightarrow \infty} \frac{S_n(x)-\max\limits_{1\leq i \leq n}a_i(x)a_{i+1}(x)}{n\log^2n}=\frac{1}{2\log2}.$$ \item The Hausdor…
