In mathematics, the Denjoy–Koksma inequality, introduced by as a combination of work of Arnaud Denjoy and the Koksma–Hlawka inequality of Jurjen Ferdinand Koksma, is a bound for Weyl sums
| m-1 | |
| \sum | |
| k=0 |
f(x+k\omega)
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
| q-1 | |
| \left|\sum | |
| i=0 |
\phi\circfi(x)-q\intT\phid\mu\right|\leqslant\operatorname{Var}(\phi)