In mathematics, the Erdős–Turán inequality bounds the distance between a

probability measure In mathematics, a probability measure is a real-valued function defined on a set of events in a σ-algebra that satisfies Measure (mathematics), measure properties such as ''countable additivity''. The difference between a probability measure an ...

on the circle and the

Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...

, in terms of Fourier coefficients. It was proved by

Paul Erdős Paul Erdős ( ; 26March 191320September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in discrete mathematics, g ...

and

Pál Turán Pál Turán (; 18 August 1910 – 26 September 1976) also known as Paul Turán, was a Hungarian mathematician who worked primarily in extremal combinatorics. In 1940, because of his Jewish origins, he was arrested by History of the Jews in Hun ...

in 1948. Let ''μ'' be a probability measure on the

unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...

R/Z. The Erdős–Turán inequality states that, for any natural number ''n'', :

\sup_A \left,  \mu(A) - \mathrm\, A \ 
     \leq C \left( \frac + \sum_^n \frac \right),

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

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

are the

Fourier coefficients A Fourier series () is an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems ...

of ''μ'', and ''C'' > 0 is a numerical constant.

Application to discrepancy

Let ''s''₁, ''s''₂, ''s''₃ ... ∈ R be a sequence. The Erdős–Turán inequality applied to the measure :

\mu_m(S) = \frac \# \, \quad S \subset discrepancy :

: \begin
D(m) & \left( = \sup_ \Big"> m^ \# \ - (b-a) \Big,  \right) \\ \sum_^m e^ \\right). 
\end \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 Low-discrepancy_sequence#The Erdős–Turán–Koksma inequality">Erdős–Turán–Koksma inequality.

Notes

Additional references

* {{DEFAULTSORT:Erdos-Turan inequality Inequalities (mathematics) Paul Erdős, Turán inequality Theorems in approximation theory