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.

The Thue–Siegel–Roth theorem says that, for algebraic irrational numbers, the exponent of 2 in the corollary to Dirichlet’s approximation theorem is the best we can do: such numbers cannot be approximated by any exponent greater than 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.