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Auxiliary Polynomials in Number Theory by David Masser

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8

Irrationality measures I – Mahler

Let θ be an irrational number, so that for any integers r and s ≥ 1 we have θr/s ≠ 0 (one usually takes p, q here, but these letters have been used up in earlier chapters). It is now natural to ask how small |θr/s| > 0 can be, and this can be answered by an inequality

image (8.1)

for all such r and s, where ϵ(r, s) > 0 is an easily calculated function only mildly dependent (if at all) on the numerical value of θ. Such a function is usually called an irrationality measure (see for example Exercise 2.5 with θ = e and We will see a very important example in Chapter 12.

If θ is not real, say with imaginary ...

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