Denjoy–Koksma inequality

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 $$\sum_{k=0}^{m-1}f(x+k\omega)$$ 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


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