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} .

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