关于Euclidean algorithms和丢番图代价的收敛速度的局部极限定理
A local limit theorem with speed of convergence for Euclidean algorithms and diophantine costs
摘要：
对于大的$N$，我们考虑$x=p/q$的普通连分式与$1\le p\le q\le N$，或者等价地，两个整数$1\le p\le q\le N$的Euclid的gcd算法，将均匀分布放在$p$和$q$的集合上。我们研究了在可能的数字集合$\mathbb{Z}_+^*$上的加性成本函数$c$的算法执行的总成本的分布，渐近地为$N\to\infty$。如果$c$是非格的，并且满足温和的增长条件，则局部极限定理已由第二作者证明。通过引入关于代价的丢番图条件，可以控制局部极限定理的收敛速度。我们使用以前的估计的第一作者和谷，我们适应我们的设置边界Dolgopyat和墨尔本的转让运营商。我们的丢番图条件是一般的（关于勒贝格测度）。对于足够光滑的观测量（取决于丢番图条件），我们达到最佳速度。
英文摘要：
For large $N$, we consider the ordinary continued fraction of $x=p/q$ with $1\le p\le q\le N$, or, equivalently, Euclid's gcd algorithm for two integers $1\le p\le q\le N$, putting the uniform distribution on the set of $p$ and $q$s. We study the distribution of the total cost of execution of the algorithm for an additive cost function $c$ on the set $\mathbb{Z}_+^*$ of possible digits, asymptotically for $N\to\infty$. If $c$ is nonlattice and satisfies mild growth conditions, the local limit theorem was proved previously by the second named author. Introducing diophantine conditions on the cost, we are able to control the speed of convergence in the local limit theorem. We use previous estimates of the first author and Vallée, and we adapt to our setting bounds of Dolgopyat and Melbourne on transfer operators. Our diophantine condition is generic (with respect to Lebesgue measure). For smooth enough observables (depending on the diophantine condition) we attain the optimal speed.
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