Denjoy–Koksma inequality

In mathematics, the Denjoy–Koksma inequality, introduced by (Herman 1979) as a combination of work of Arnaud Denjoy and the Koksma–Hlawka inequality of Jurjen Ferdinand Koksma, is a bound for Weyl sums [math]\displaystyle{ \sum_{k=0}^{m-1}f(x+k\omega) }[/math] of functions f of bounded variation.

Statement

Suppose that a map f from the circle T to itself has irrational rotation number α, and p/q is a rational approximation to α with p and q coprime, |α – p/q| < 1/q². Suppose that φ is a function of bounded variation, and μ a probability measure on the circle invariant under f. Then

[math]\displaystyle{ \left|\sum_{i=0}^{q-1} \phi \circ f^i(x) - q\int_T \phi \, d\mu \right| \leqslant \operatorname{Var}(\phi) }[/math]

(Herman 1979)

References

• Herman, Michael-Robert (1979), "Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations", Publications Mathématiques de l'IHÉS (49): 5–233, ISSN 1618-1913, http://www.numdam.org/item?id=PMIHES_1979__49__5_0 
• Kuipers, L.; Niederreiter, H. (1974), Uniform distribution of sequences, New York: Wiley-Interscience [John Wiley & Sons], Reprinted by Dover 2006, ISBN 978-0-486-45019-3