Extensions of the Cugiani-Mahler theorem

Abstract

In 1955, Roth established that if ξ is an irrational number such that there are a positive real number ε and infinitely many rational numbers p/q with q ≥ 1 and |ξ − p/q| < q −2−ε , then ξ is transcendental. A few years later, Cugiani obtained the same conclusion with ε replaced by a function q → ε(q) that decreases very slowly to zero, provided that the sequence of rational solutions to |ξ − p/q| < q −2−ε(q) is sufficiently dense, in a suitable sense. We give an alternative, and much simpler, proof of Cugiani's Theorem and extend it to simultaneous approximation.

On two notions of complexity of algebraic numbers

“…We deduce from Corollary 5.2 an improvement of an extension due to Mahler [17] of a theorem of Cugiani [11], see [9] for further references on the Cugiani-Mahler Theorem.…”

“…Arguing as in [9], we get the following improvement of Theorem 1 on page 169 of [17], that we state without proof. For a positive integer m, we denote by exp m the mth iterate of the exponential function and by log m the function that coincides with the mth iterate of the logarithm function on [exp m 1, +∞) and that takes the value 1 on (−∞, exp m 1].…”

“…By combining ideas from [9] with a suitable version of the Quantitative Subspace Theorem, we are able to prove the following concerning (2.1).…”

“…Ideas from [9] combined with Theorem 3.1 from [14] allow us to prove a weaker version of Theorem 2.1, namely to conclude that (2.2) holds for any η smaller than 1/(4ω(b) + 15). The key point for removing the dependence on b is the use of Theorem 5.1 below, and more precisely the fact that the exponent on ε −1 in (5.9) does not depend of the cardinality of the set of places S.…”

A further improvement of the Quantitative Subspace Theorem

“…As it turned out, this version is in general more useful for applications than the existing quantitative versions of the basic Subspace Theorem concerning (1.1). For instance, the work of Evertse and Schlickewei led to uniform upper bounds for the number of solutions of linear equations in unknowns from a multiplicative group of finite rank [12] and for the zero multiplicity of linear recurrence sequences [27], and more recently to results on the complexity of b-ary expansions of algebraic numbers [6], [3], to improvements and generalizations of the Cugiani-Mahler theorem [2], and approximation to algebraic numbers by algebraic numbers [5]. For an overview of recent applications of the Quantitative Subspace Theorem we refer to Bugeaud's survey paper [4].…”