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

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

Here ${\displaystyle \lfloor N\rfloor }$ 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

${\displaystyle 0<\left|\alpha -{\frac {p}{q}}\right|<{\frac {1}{q^{2}}}}$

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.