Diophantine Approximation

Diophantine approximation studies how well real numbers can be approximated by rational numbers. A central question is: how small can

[ \left|\alpha - \frac{p}{q}\right| ]

be, in terms of the denominator (q)?

Dirichlet's approximation theorem

For any real number (\alpha) and any integer (N>1), there exist integers (p,q) with (1 \le q < N) such that

[ \left|\alpha - \frac{p}{q}\right| < \frac{1}{qN} \le \frac{1}{q^2}. ]

In particular, every irrational (\alpha) has infinitely many rational approximations (p/q) satisfying

[ \left|\alpha - \frac{p}{q}\right| < \frac{1}{q^2}. ]

Continued fractions

The “best” rational approximations to (\alpha) come from its continued fraction convergents.

If (\alpha = [a_0; a_1, a_2, \dots]) and (p_n/q_n = [a_0; a_1, \dots, a_n]) is the (n)-th convergent, then

[ \left|\alpha - \frac{p_n}{q_n}\right| < \frac{1}{q_n q_{n+1}} < \frac{1}{q_n^2}. ]

Limits for algebraic numbers

Liouville's theorem gives a first limitation: if (\alpha) is algebraic of degree (d>1), then there exists (c(\alpha)>0) such that for all rationals (p/q),

[ \left|\alpha - \frac{p}{q}\right| > \frac{c(\alpha)}{q^d}. ]

Roth's theorem is a deep strengthening: for any algebraic (\alpha) and any (\varepsilon>0), the inequality

[ \left|\alpha - \frac{p}{q}\right| < \frac{1}{q^{2+\varepsilon}} ]

has only finitely many solutions in coprime integers (p,q). The exponent (2) is essentially best possible by Dirichlet's theorem.