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Questions related to various forms of integration including the Riemann integral, Lebesgue integral, Riemann–Stieltjes integral, double integrals, line integrals, contour integrals, surface integrals, integrals of differential forms, ...

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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 …
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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 $ …
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