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 shows that any irrational number has irrationality measure at least 2. 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. The Thue–Siegel–Roth theorem uses advanced techniques of number theory, but many simpler numbers such as the golden ratio ${\displaystyle (1+{\sqrt {5}})/2}$ can be much more easily verified to be inapproximable beyond exponent 2. This exponent is referred to as the irrationality measure.

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

Proof By The Pigeonhole Principle

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]

Proof Outline: Let ${\displaystyle \alpha }$ be an irrational number and ${\displaystyle n}$ be an integer. For every ${\displaystyle k=0,1,...,n}$ we can write ${\displaystyle k\alpha =m_{k}+x_{k}}$ such that ${\displaystyle m_{k}}$ is an integer and ${\displaystyle 0\leq x_{k}<1}$. One can divide the interval ${\displaystyle [0,1)}$ into ${\displaystyle n}$ smaller intervals of measure ${\displaystyle {\frac {1}{n}}}$. Now, we have ${\displaystyle n+1}$ numbers ${\displaystyle x_{0},x_{1},...,x_{n}}$ and ${\displaystyle n}$ intervals. Therefore, by the pigeonhole principle, at least two of them are in the same interval. We can call those ${\displaystyle x_{i},x_{j}}$ such that ${\displaystyle i. Now:

${\displaystyle |(j-i)\alpha -(m_{j}-m_{i})|=|j\alpha -m_{j}-(i\alpha -m_{i})|=|x_{j}-x_{i}|<{\frac {1}{n}}}$

Dividing both sides by ${\displaystyle j-i}$ will result in:

${\displaystyle \left|\alpha -{\frac {m_{j}-m_{i}}{j-i}}\right|<{\frac {1}{(j-i)n}}\leq {\frac {1}{\left(j-i\right)^{2}}}}$

And we proved the theorem.

Proof By Minkowski's theorem

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

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

Notes

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 ^ http://jeff560.tripod.com/p.html for a number of historical references.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