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
Jurjen Ferdinand Koksma (21 April 1904, Schoterland – 17 December 1964, Amsterdam) was a Dutch mathematician who specialized in analytic number theory.
Koksma received his Ph.D. degree (''cum laude'') in 1930 at the University of Gronin ...
, is a bound for
Weyl sum
In mathematics, an exponential sum may be a finite Fourier series (i.e. a trigonometric polynomial), or other finite sum formed using the exponential function, usually expressed by means of the function
:e(x) = \exp(2\pi ix).\,
Therefore, a typic ...
s
of functions ''f'' of
bounded variation
In mathematical analysis, a function of bounded variation, also known as ' function, is a real number, real-valued function (mathematics), function whose total variation is bounded (finite): the graph of a function having this property is well beh ...
.
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
In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equiv ...
, , ''α'' – ''p''/''q'', < 1/''q''
2. Suppose that ''φ'' is a function of bounded variation, and ''μ'' a
probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a σ-algebra that satisfies Measure (mathematics), measure properties such as ''countable additivity''. The difference between a probability measure an ...
on the circle invariant under ''f''. Then
:
References
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{{DEFAULTSORT:Denjoy-Koksma inequality
Theorems in mathematical analysis
Inequalities (mathematics)