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Auxiliary Polynomials in Number Theory by David Masser

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12

Irrationality measures II – Thue

The irrationality of algebraic numbers like image or image is rather trivial, and so we can jump straight to the problem of irrationality measures in the sense of Chapter 8. One argument is fast, using the notation of the previous chapter. Namely if image is near r/s (with r in Z and s in N), then ω = r is small and as soon as |ω| ≤ 1 we get |r| ≤ (α+1)s so ||ω|| ≤ |r|+s||α|| ≤ (2α+1)s. Thus by (11.9) with d = 2 we get

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