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,

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

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:

$\begin{align} D(m) & \left( = \sup_{0 \leq a \leq b \leq 1} \Big| m^{-1} \# \{ 1 \leq j \leq m \, | \, a \leq s_j \, \mathrm{mod} \, 1 \leq b \} - (b-a) \Big| \right) \\[8pt] & \leq C \left( \frac{1}{n} + \frac{1}{m} \sum_{k=1}^n \frac{1}{k} \left| \sum_{j=1}^m e^{2 \pi i s_j k} \right|\right). \end{align} \qquad (1)$

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.

Notes

1. 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.
2. 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.

Additional references

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