Denjoy–Koksma inequality

In mathematics, the Denjoy–Koksma inequality, introduced by Herman (1979, p.73) as a combination of work of Arnaud Denjoy and the Koksma–Hlawka inequality of Jurjen Ferdinand Koksma, is a bound for Weyl sums \sum_{k=0}^{m-1}f(x+k\omega) 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

\left|\sum_{i=0}^{q-1} \phi f^i(x) - q\int_T \phi \, d\mu \right| < \operatorname{Var}(\phi)

(Herman 1979, p.73)

References

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