Erdős–Turán inequality - Alchetron, The Free Social Encyclopedia

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,

sup A | μ ( A ) − m e s A | ≤ C ( 1 n + ∑ k = 1 n | μ ^ ( k ) | k ) ,

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

μ ^ ( k ) = ∫ exp ⁡ ( 2 π i k θ ) d μ ( θ )

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

μ m ( S ) = 1 m # { 1 ≤ j ≤ m | s j m o d 1 ∈ S } , S ⊂ [ 0 , 1 ) ,

yields the following bound for the discrepancy:

D ( m ) ( = sup 0 ≤ a ≤ b ≤ 1 | m − 1 # { 1 ≤ j ≤ m | a ≤ s j m o d 1 ≤ b } − ( b − a ) | ) ≤ C ( 1 n + 1 m ∑ k = 1 n 1 k | ∑ j = 1 m e 2 π i s j k | ) . ( 1 )

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.

References

Erdős–Turán inequality Wikipedia

(Text) CC BY-SA