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.^[1]^[2]

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.

↑ Erdős, P.; Turán, P. (1948). "On a problem in the theory of uniform distribution. I.". Nederl. Akad. Wetensch. 51: 1146–1154. MR 0027895. Zbl 0031.25402.
↑ Erdős, P.; Turán, P. (1948). "On a problem in the theory of uniform distribution. II.". Nederl. Akad. Wetensch. 51: 1262–1269. MR 0027895. Zbl 0032.01601.

Harman, Glyn (1998). Metric number theory. London Mathematical Society Monographs. New Series. 18. Oxford: Clarendon Press. ISBN 0-19-850083-1. Zbl 1081.11057.

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