Hurwitz's theorem (number theory)

Remarks

The proof uses the continued fraction expansion of $\xi$. Among any three consecutive convergents $p_n/q_n$ of $\xi$, at least one satisfies the inequality, and there are infinitely many convergents. For the second part, take the golden ratio $\varphi = (1 + \sqrt{5})/2$, whose continued fraction is $[1; 1, 1, 1, \dots]$. Its convergents are quotients of consecutive Fibonacci numbers, and $q^2 \abs{\varphi - p/q}$ tends to $1/\sqrt{5}$ along them. The theorem improves the constant $1$ of Dirichlet's approximation theorem. If one removes $\varphi$ and all numbers equivalent to it under the action of $\mathrm{GL}_2(\Z)$, the best constant becomes $\sqrt{8}$.

Going deeper

For a student

The theorem belongs to the theory of the Lagrange spectrum, which records the best constants for each irrational number. Continued fractions are the main tool: the convergents give the best approximations. Numbers with bounded partial quotients, such as quadratic irrationals, are badly approximable, and the golden ratio is the extreme case.