Irrationality exponent and rational approximations with prescribed growth

Irrationality exponent and rational approximations with prescribed growth Stephane Fischler and Tanguy Rivoal June 10, 2009 1 Introduction In 1978, Apery [2] proved the irrationality of ?(3) by constructing two explicit sequences of integers (un)n and (vn)n such that 0 6= un?(3)? vn ? 0 and un ? +∞, both at geometric rates. He also deduced from this an upper bound for the irrationality exponent µ(?(3)) of ?(3). In general, the irrationality exponent µ(?) of an irrational number ? is defined as the infimum of all real numbers µ such that the inequality ????? ? p q ???? > 1 qµ holds for all integers p, q, with q sufficiently large. It is well-known that µ(?) ≥ 2 for any irrational number ? and that it equals 2 for almost all irrational numbers. After Apery, the following lemma has often been used to bound µ(?) from above, for example for the numbers log(2) and ?(2). (Other lemmas can be used to bound the irrationality exponent of numbers of a different nature, like exp(1).) Lemma 1. Let ? ? R \ Q, and ?, ? be real numbers such that 0 < ? < 1 and ? > 1.

• ??? ≤


• coprime integers


• numbers


• all ?


• answers completely all


• vn


• there exist


• sequences


• un? ?