6

Diophantine approximation

**6.1 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 |*q**θ* − *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*}, {2*θ*}, …, {(*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 ...

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