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 s1, s2, s3 ... ∈ R be a sequence. The Erdős–Turán inequality applied to the measure
yields the following bound for the discrepancy:
A multi-dimensional variant of (1) is known as the Erdős–Turán–Koksma inequality.
- Erdős, P.; Turán, P. (1948). "On a problem in the theory of uniform distribution. I.". Nederl. Akad. Wetensch. 51: 1146–1154. MR 0027895. Zbl 0031.25402.
- Erdős, P.; Turán, P. (1948). "On a problem in the theory of uniform distribution. II.". Nederl. Akad. Wetensch. 51: 1262–1269. MR 0027895. Zbl 0032.01601.