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. Application to discrepancy . Let "s", "s", "s" ... ∈ 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 (1) is known as the Erdős–Turán–Koksma inequality.