Erdős–Turán inequality

In mathematics, the Erdős–Turán inequality bounds the distance between a probability measure on the circle and the Lebesgue measure, in terms of Fourier coefficients. It was proved by Paul Erdős and Pál Turán in 1948.

Let μ be a probability measure on the unit circle R/Z. The Erdős–Turán inequality states that, for any natural number n,

$\sup _{A}\left|\mu (A)-\mathrm {mes} \,A\right|\leq C\left({\frac {1}{n}}+\sum _{k=1}^{n}{\frac {|{\hat {\mu }}(k)|}{k}}\right),$

where the supremum is over all arcs A ⊂ R/Z of the unit circle, mes stands for the Lebesgue measure,

${\hat {\mu }}(k)=\int \exp(2\pi ik\theta )\,d\mu (\theta )$

are the Fourier coefficients of μ, and C > 0 is a numerical constant.

Application to discrepancy

Let s₁, s₂, s₃ ... ∈ R be a sequence. The Erdős–Turán inequality applied to the measure

$\mu _{m}(S)={\frac {1}{m}}\#\{1\leq j\leq m\,|\,s_{j}\,\mathrm {mod} \,1\in S\},\quad S\subset [0,1),$

yields the following bound for the discrepancy:

${\begin{aligned}D(m)&\left(=\sup _{0\leq a\leq b\leq 1}{\Big |}m^{-1}\#\{1\leq j\leq m\,|\,a\leq s_{j}\,\mathrm {mod} \,1\leq b\}-(b-a){\Big |}\right)\\[8pt]&\leq C\left({\frac {1}{n}}+{\frac {1}{m}}\sum _{k=1}^{n}{\frac {1}{k}}\left|\sum _{j=1}^{m}e^{2\pi is_{j}k}\right|\right).\end{aligned}}\qquad (1)$

This inequality holds for arbitrary natural numbers m,n, and gives a quantitative form of Weyl's criterion for equidistribution.

A multi-dimensional variant of (1) is known as the Erdős–Turán–Koksma inequality.

Additional references

• Harman, Glyn (1998). Metric Number Theory. London Mathematical Society Monographs. New Series. Vol. 18. Clarendon Press. ISBN 0-19-850083-1. Zbl 1081.11057.