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user 27637
7
votes
Accepted
Speed of convergence for Weyl's Equidistribution theorem
As a complement to Gerry Myerson answer, you can bound the discrepancy $D_N$ using the Erdos-Turan inequality
$$
D_{N} \leq \frac{\log 2}{\pi (H + 1)} + \frac{1}{\pi N} \sum_{h = 1}^{H} \frac{1}{h}
\b …
- 166
5
votes
Speed of convergence for Weyl's Equidistribution theorem
As you see from the above two answers the rate of convergence will depend on the diophantine nature of $\alpha$. Indeed
$$
\sum_{n = 1}^{N} \exp(2\pi i h \alpha n)\ll \min(N,1/||h \alpha||)
$$
where $ …
- 166