Large deviations for denominators of continued fractions
Large deviations for denominators of continued fractions
复制标题
DOI：
10.1088/1361-6544/ab9a1d
复制
发表时间：
2019-04
期刊：
影响因子：
1.7
通讯作者：
Hiroki Takahasi
中科院分区：
文献类型：
--
作者：
Hiroki Takahasi
我们给出了正则连分式展开中第 n 次收敛的分母远离均值 nπ212log2 的概率的指数上限。指数率是最好的，由与高斯变换的李雅普诺夫指数维数谱相关的解析函数给出。我们还建立了分母的大偏差原理（LDP）。作为推论，我们推导了周期连分数和连分数原像的分母的 LDP。主要结果的证明依赖于有限拓扑马尔可夫位移的热力学形式。
We give exponential upper bounds on the probability with which the denominator of the nth convergent in the regular continued fraction expansion stays away from the mean nπ212log2. The exponential rate is best possible, given by an analytic function related to the dimension spectrum of Lyapunov exponents for the Gauss transformation. We also establish the large deviation principle (LDP) for denominators. As corollaries, we derive the LDPs for denominators of periodic continued fractions and continued fraction preimages. Proofs of the main results rely on the thermodynamic formalism for finite topological Markov shifts.