Dirichlet's approximation theorem

In number theory, Dirichlet's theorem on Diophantine approximation, also called Dirichlet's approximation theorem, states that for any real numbers ${\displaystyle \alpha }$ and ${\displaystyle N}$, with ${\displaystyle 1\leq N}$, there exist integers ${\displaystyle p}$ and ${\displaystyle q}$ such that ${\displaystyle 1\leq q\leq N}$ and

${\displaystyle \left|q\alpha -p\right|\leq {\frac {1}{[N]+1}}<{\frac {1}{N}}.}$

Here ${\displaystyle [N]}$ represents the integer part of ${\displaystyle N}$. This is a fundamental result in Diophantine approximation, showing that any real number has a sequence of good rational approximations: in fact an immediate consequence is that for a given irrational α, the inequality

${\displaystyle \left|\alpha -{\frac {p}{q}}\right|<{\frac {1}{q^{2}}}}$

is satisfied by infinitely many integers p and q. This corollary also shows that the Thue–Siegel–Roth theorem, a result in the other direction, provides essentially the tightest possible bound, in the sense that the bound on rational approximation of algebraic numbers cannot be improved by increasing the exponent beyond 2.

Simultaneous version

The simultaneous version of the Dirichlet's approximation theorem states that given real numbers ${\displaystyle \alpha _{1},\ldots ,\alpha _{d}}$ and a natural number ${\displaystyle N}$ then there are integers ${\displaystyle p_{1},\ldots ,p_{d},q\in \mathbb {Z} ,1\leq q\leq N}$ such that ${\displaystyle \left|\alpha _{i}-{\frac {p_{i}}{q}}\right|\leq {\frac {1}{qN^{1/d}}}.}$

Method of proof

This theorem is a consequence of the pigeonhole principle. Peter Gustav Lejeune Dirichlet who proved the result used the same principle in other contexts (for example, the Pell equation) and by naming the principle (in German) popularized its use, though its status in textbook terms comes later.[1] The method extends to simultaneous approximation.[2]

Another simple proof of the Dirichlet's approximation theorem is based on Minkowski's theorem applied to the set

${\displaystyle S=\left\{(x,y)\in \mathbb {R} ^{2};-N-{\frac {1}{2}}\leq x\leq N+{\frac {1}{2}},\vert \alpha x-y\vert \leq {\frac {1}{N}}\right\}.}$

Since the volume of ${\displaystyle S}$ is greater than ${\displaystyle 4}$, Minkowski's theorem establishes the existence of a non-trivial point with integral coordinates. This proof extends naturally to simultaneous approximations by considering the set

${\displaystyle S=\left\{(x,y_{1},\dots ,y_{d})\in \mathbb {R} ^{1+d};-N-{\frac {1}{2}}\leq x\leq N+{\frac {1}{2}},|\alpha _{i}x-y_{i}|\leq {\frac {1}{N^{1/d}}}\right\}.}$

See also

• Dirichlet's theorem on arithmetic progressions
• Hurwitz's theorem (number theory)
• Heilbronn set
• Kronecker's theorem (generalization of Dirichlet's theorem)

Notes

1. http://jeff560.tripod.com/p.html for a number of historical references.
2. "Dirichlet theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994]

References

• Schmidt, Wolfgang M (1980). Diophantine approximation. Lecture Notes in Mathematics. 785. Springer. doi:10.1007/978-3-540-38645-2. ISBN 978-3-540-38645-2.
• Schmidt, Wolfgang M. (1991). Diophantine Approximations and Diophantine Equations. Lecture Notes in Mathematics book series. 1467. Springer. doi:10.1007/BFb0098246. ISBN 978-3-540-47374-9.

External links

• Dirichlet's Approximation Theorem at PlanetMath.