Dirichlet's theorem

Diophantine approximation is concerned with the solubility of inequalities in integers. The simplest result in this field was obtained by Dirichlet in 1842. He showed that, for any real θ and any integer Q > 1 there exist integers p, q with 0 < q < Q such that |qθ − p| ≤ 1/Q.

The result can be derived at once from the so-called ‘box’ or ‘pigeon-hole’ principle. This asserts that if there are n holes containing n + 1 pigeons then there must be at least two pigeons in some hole. Consider in fact the Q + 1 numbers 0, 1, {θ}, {2θ}, …, {(Q − 1)θ}, where {x} denotes the fractional part of x as in Chapter 2. These numbers all lie in the interval [0, 1], and if one divides the latter, as clearly one can, into Q disjoint sub-intervals, each of length 1/Q, then it follows that two of the Q + 1 numbers must lie in one of the Q sub-intervals. The difference between the two numbers has the form qθ − p, where p, q are integers with 0 < q < Q, and we have |qθ − p| ≤ 1/Q, as required.

Dirichlet's theorem holds more generally for any real Q > 1; the result for non-integral Q follows from the theorem just established with Q replaced by [Q] + 1. Further it is clear that the integers p, q referred to in the theorem can be chosen to be relatively prime.