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 AR/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[edit]

Let s1, s2, s3 ... ∈ 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.

Notes[edit]

  1. ^ 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. 
  2. ^ 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. 

Additional references[edit]