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,

${\displaystyle \sup _{A}\left|\mu (A)-\mathrm {mes} \,A\right|\leq C\left({\frac {1}{n}}+\sum _{k=1}^{n}{\frac {|{\hat {\mu }}(k)|}{k}}\right),}$

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

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

are the Fourier coefficients of μ, and C > 0 is a numerical constant.