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, 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.
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.