Denjoy–Koksma inequality

In mathematics, the Denjoy–Koksma inequality, introduced by Template:Harvtxt 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

$|\sum _{i=0}^{q-1}\varphi \circ f^i(x)-q\underset{T}{\int }\varphi d\mu |⩽\mathrm{Var}(\varphi )$

References

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