Given a real number α∈[0,1]\alpha\in \lbrack 0,1\rbrack and an integer Q>0Q>0, Dirichlet’s approximation theorem guarantees the existence of a rational a/qa/q, q≤Qq\leq Q, such that

|α−aq|≤1qQ.\left|\alpha – \frac{a}{q}\right|\leq \frac{1}{q Q}.

Question: how do we find such a q≤Qq\leq Q?

Continued fractions give answers to related but slightly different questions (finding qq such that |α−a/q|≤1/qQ|\alpha – a/q|\leq 1/q Q; finding best approximants). Can we use continued fractions to find a rational a/qa/q answering my question above?

(I suppose so, but I’d like to know how! Perhaps it is obvious.)

(Yes, I know, I should have learnt this in elementary school.)

=================

Yes, there is a theorem which states that the best approximant for a given QQ is given by some approximant in the continued fraction expansion. See for example Hardy and Wright 3ed. for a discussion of this.

– Adam Hughes

2 days ago

a clear short document, among others (math.harvard.edu/archive/118r_spring_05/handouts/…)

– JeanMarie

2 days ago

On JeanMarie’s answer: no, that does not answer the question above!

– H A Helfgott

2 days ago

However, Them 164 in Hardy-Wright does (and so does Them 9 in Khinchin). The best-approximate property is neither needed nor immediately useful.

– H A Helfgott

yesterday

=================

=================