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 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/q''2. Suppose that φ is a function of bounded variation, and μ a probability measure on the circle invariant under f. Then