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 ∑ k = 0 m − 1 f ( x + k ω ) {\displaystyle \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
| ∑ i = 0 q − 1 ϕ ∘ f i ( x ) − q ∫ T ϕ d μ | ⩽ Var ( ϕ ) {\displaystyle \left|\sum _{i=0}^{q-1}\phi \circ f^{i}(x)-q\int _{T}\phi \,d\mu \right|\leqslant \operatorname {Var} (\phi )}
(Herman 1979, p.73)
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, MR 0538680 Kuipers, L.; Niederreiter, H. (1974), Uniform distribution of sequences, New York: Wiley-Interscience [John Wiley & Sons], ISBN 978-0-486-45019-3, MR 0419394, Reprinted by Dover 2006