Restricted rational approximation and Apéry-type constructions

Abstract

Let ξ be a real irrational number, and φ be a function (satisfying some assumptions). In this text we study the φ-exponenl of irrationality of ξ, defined as the supremum of the set of μ for which there are infinitely many q ≥ 1 such that q is a multiple of φ(q) and | ξ − p q | ≤ q − u for some p ∈ ℤ. We obtain general results on this exponent (a lower bound, the Haussdorff dimension of the set where it is large,…) and connect it to sequences of small linear forms in 1 and ξ with integer coefficients, with geometric behaviour and a divisibility property of the coefficients. Using Apéry's proof that ζ(3) is irrational, we obtain an upper bound for the φ-exponent of irrationality of ζ (3), for a given φ.