title: Denjoy–Koksma inequality
text: # Denjoy–Koksma inequality
  
  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 $\sum _{k=0}^{m-1}f(x+k\omega )$ 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| < $1/q^2.$ Suppose that φ is a function of bounded variation, and μ a probability measure on the circle invariant under f. Then
  
      $\left|\sum _{i=0}^{q-1}\phi \circ f^{i}(x)-q\int _{T}\phi \,d\mu \right|\leqslant \operatorname {Var} (\phi )$
  
  (Herman 1979, p.73)
  
  