旋转不可积函数Birkoff和的一个极限定理
A Limit Theorem for Birkoff sums of non-integrable functions over rotations
摘要：
本文考虑旋转上具有1/x型奇点的函数的Birkhoff和，证明了下面的极限定理.令$S_N= S_N(α,x)$为第N阶非重整化Birkhoff和，其中$x in [0,1)$为初始点，$α\in [0,1)$为旋转数，并且$(α, x)$均匀分布。证明了当N趋于无穷大时，$S_N/N$在$(α,x)$中有一个联合极限分布.作为推论，我们得到了某些三角和的极限分布的存在性。
英文摘要：
We consider Birkhoff sums of functions with a singularity of type 1/x over rotations and prove the following limit theorem. Let $S_N= S_N(α,x)$ be the N^th non-renormalized Birkhoff sum, where $x in [0,1)$ is the initial point, $α\in [0,1)$ is the rotation number and $(α, x)$ are uniformly distributed. We prove that $S_N/N$ has a joint limiting distribution in $(α,x)$ as N tends to infinity. As a corollary, we get the existence of a limiting distribution for certain trigonometric sums.
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