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,
where the supremum is over all arcs A ⊂ R/Z of the unit circle, mes stands for the Lebesgue measure,
are the Fourier coefficients of μ, and C > 0 is a numerical constant.Let s1, s2, s3... ∈ R be a sequence. The Erdős–Turán inequality applied to the measure
yields the following bound for the discrepancy:
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 is known as the Erdős–Turán–Koksma inequality.
