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Weyl sum and the ergodicity of the shift map on $T$
Weyl Sum The Weyl sum, an object of central importance in analytic number theory, is defined as follows: $$S(N, \alpha) = \sum_{n=1}^{N} e^{2\pi i f(n) \alpha}$$ where $N$ is a positive integer and $\alpha$ is a real number. The behavior of this sum as $N$ grows large is of particular interest. The size of the Weyl sum is often related to the behavior of $f(n)$. Generally, if we impose some regularity on $f(n)$, such as continuous derivatives, Lipschitz continuity, or continuous second derivativ…
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Several important articles in the investigation of the equidistribution problem
In this blog, we mark some article that is important in the investigation of the equidistribution problem, each of them providing new inspirit to solve some particuler type of equidistribution problem. 1. Mixing, counting, and equidistribution in Lie groups The author is Alex Eskin, Curt McMullen This article was published in 1993. in Duke Math. J. 71(1): 181-209 (July 1993). DOI: 10.1215/S0012-7094-93-07108-6 Let $\Gamma \subset G=\operatorname{Aut}\left(\mathbb{H}^2\right)$ be a group of isome…