论文标题
关于用固定分母减少分数的部分商的分布
On the distribution of partial quotients of reduced fractions with fixed denominator
论文作者
论文摘要
在本文中，我们研究了部分商$ a/n $的部分分数扩展中部分商的分配属性，其中$ n $是固定的，$ a $ a $通过Mod $ n $ bistion类，与$ n $相关。我们的方法涵盖了统计数据，例如部分商的总和，最大的部分商，部分商的经验分布，Dedekind Sums等等。我们证明了部分商的总和，对最大部分商和Dedekind总和的尖锐尾巴估算呈尖锐的浓度不平等，所有这些都匹配了极限定律中的尾巴行为，这些行为是在一组可能的分母$ n $的额外平均值中所知的。我们表明，用固定分母减少分数的局部分数的分布非常适合高斯-Kuzmin分布。作为推论，我们建立了减少部分的存在，而少量的部分商的resp。小小的最大部分商。
In this paper, we study distributional properties of the sequence of partial quotients in the continued fraction expansion of fractions $a/N$, where $N$ is fixed and $a$ runs through the set of mod $N$ residue classes which are coprime with $N$. Our methods cover statistics such as the sum of partial quotients, the maximal partial quotient, the empirical distribution of partial quotients, Dedekind sums, and much more. We prove a sharp concentration inequality for the sum of partial quotients, and sharp tail estimates for the maximal partial quotient and for Dedekind sums, all matching the tail behavior in the limit laws which are known under an extra averaging over the set of possible denominators $N$. We show that the distribution of partial quotients of reduced fractions with fixed denominator gives a very good fit to the Gauss-Kuzmin distribution. As corollaries we establish the existence of reduced fractions with a small sum of partial quotients resp. a small maximal partial quotient.
