Editing Diophantine approximation (section)

== Uniform distribution ==
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Another topic that has seen a thorough development is the theory of [[equidistributed sequence|uniform distribution mod 1]]. Take a sequence ''a''<sub>1</sub>, ''a''<sub>2</sub>, ... of real numbers and consider their ''fractional parts''. That is, more abstractly, look at the sequence in <math>\mathbb{R}/\mathbb{Z}</math>, which is a circle. For any interval ''I'' on the circle we look at the proportion of the sequence's elements that lie in it, up to some integer ''N'', and compare it to the proportion of the circumference occupied by ''I''. ''Uniform distribution'' means that in the limit, as ''N'' grows, the proportion of hits on the interval tends to the 'expected' value. [[Hermann Weyl]] proved a [[Weyl's criterion|basic result]] showing that this was equivalent to bounds for exponential sums formed from the sequence. This showed that Diophantine approximation results were closely related to the general problem of cancellation in exponential sums, which occurs throughout [[analytic number theory]] in the bounding of error terms.

Related to uniform distribution is the topic of [[irregularities of distribution]], which is of a [[combinatorics|combinatorial]] nature.