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Irrationality exponent and rational approximations with prescribed growth

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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? ?


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Irrationality
exponent and rational approximations with prescribed growth
St´ephaneFischlerandTanguyRivoal
Introduction
June 10, 2009
In1978,Ape´ry[2]provedtheirrationalityofζ(3) by constructing two explicit sequences of integers (un)nand (vn)nsuch that 06=unζ(3)vn0 andun+, both at geometric rates. He also deduced from this an upper bound for the irrationality exponentµ(ζ(3)) of ζ(3). In general, the irrationality exponentµ(ξ) of an irrational numberξis defined as the infimum of all real numbersµsuch that the inequality p1 ξ> ¯ ¯µ q q holds for all integersp, q, withqsufficiently large. It is well-known thatµ(ξ)2 for any irrational numberξlnnabeum.Arserfte´pA,yrdnathatitequals2forlaomtslailrrtaoi the following lemma has often been used to boundµ(ξ) from above, for example for the numbers log(2) andζ(2). (Other lemmas can be used to bound the irrationality exponent of numbers of a different nature, like exp(1).) Lemma 1.LetξR\Q, andα, βbe real numbers such that0< α <1andβ >1. Assume there exist integer sequences(un)n1and(vn)n1such that 1/n1/n lim|unξvn|=αandlim sup|un| ≤β.(1.1) n+n+logβ Then we haveµ(ξ)1. logα The proof of Lemma 1 is not difficult. Many variants of this result exist; a slightly more general version of Lemma 1 will be proved in§4.1. Another variant, proved in [6] (Proposition 3.1), asserts that Lemma 1 holds when (1.1) is replaced with |un+1ξvn+1|un+1 lim supαsupand lim β. n+|unξvn|n+un
In this text, we prove that Lemma 1 and these variants are best possible, by obtaining a very precise converse result:
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