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,


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

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.