Erdős–Turán inequality
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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.