Dirichlet’s approximation theorem
Theorem (Dirichlet, c. 1840): For any real number and any integer , there exist integers and such that and .
Proof: We may assume .
For each integer in the interval , write
, where denotes
the greatest integer less than . Since the
numbers all lie in the same unit interval, some two
of them differ (in absolute value
) by at most .
If or is in any such pair, then the other element of the
pair is one of the , and we are done.
If not, then for some distinct
and . If we have , since each side is in
and the difference
between them is an integer. Similarly,
if , we have . So, with or
respectively, we get
where is or , and the result follows.
It is clear that we can add the condition to the conclusion.
The same statement, but with the weaker conclusion
,
admits a slightly shorter proof, and is sometimes also referred to
as the Dirichlet approximation theorem. (It was that shorter proof
which made the “pigeonhole principle
” famous.) Also, the theorem
is sometimes restricted to irrational values of , with the
(nominally stronger) conclusion
.
Title | Dirichlet’s approximation theorem |
---|---|
Canonical name | DirichletsApproximationTheorem |
Date of creation | 2013-03-22 13:15:37 |
Last modified on | 2013-03-22 13:15:37 |
Owner | Koro (127) |
Last modified by | Koro (127) |
Numerical id | 7 |
Author | Koro (127) |
Entry type | Theorem |
Classification | msc 11J04 |
Synonym | Dirichlet approximation theorem |
Related topic | IrrationalityMeasure |