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 AR/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

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