Irrationality Measures, Irrationality Bases, and a Theorem of Jarnik

Details Irrationality Measures, Irrationality Bases, and a Theorem of Jarnik Irrationality Measures, Irrationality Bases, and a Theorem of Jarnik

2004-06-15

arxiv.org/abs/math/0406300v1

Mathematics

Number Theory

14 pages, presented in part at Journe\'es Arithme\'tiques XXIII in Graz

Scientific paper

In math.NT/0307308 we defined the irrationality base of an irrational number and, assuming a stronger hypothesis than the irrationality of Euler's constant, gave a conditional upper bound on its irrationality base. Here we develop the general theory of the irrationality exponent and base, giving formulas and bounds for them using continued fractions and the Fibonacci sequence. A theorem of Jarnik on Diophantine approximation yields numbers with prescribed irrationality measure. By another method we explicitly construct series with prescribed irrationality base. Many examples are given.

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