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 α {\displaystyle \alpha } and N {\displaystyle N} , with 1 ≤ N {\displaystyle 1\leq N} , there exist integers p {\displaystyle p} and q {\displaystyle q} such that 1 ≤ q ≤ N {\displaystyle 1\leq q\leq N} and | q α − p | ≤ 1 ⌊ N ⌋ + 1 < 1 N . {\displaystyle \left|q\alpha -p\right|\leq {\frac {1}{\lfloor N\rfloor +1}}<{\frac {1}{N}}.} Here ⌊ N ⌋ {\displaystyle \lfloor N\rfloor } represents the integer part of N {\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 | α − p q | < 1 q 2 {\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 exponent 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 ( 1 + 5 ) / 2 {\displaystyle (1+{\sqrt {5}})/2} can be much more easily verified to be inapproximable beyond exponent 2.