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 )}
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/q 2 . 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 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 
