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A Comprehensive Course in Number Theory by Alan Baker

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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 |qθp| ≤ 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 ...

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