The tau_n=n conjecture in approximation by algebraic integers is false

Number theory 1 Minute

You need to know: Polynomial, degree of a polynomial, leading coefficient, irreducible polynomial over the integers, supremum.

Background: A real number \alpha is called algebraic number if there exists a polynomial P with integer coefficients such that P(\alpha)=0. Real numbers which are not algebraic are called transcendental. Below, we may and will assume that P is irreducible over the integers. The largest absolute value of the coefficients of P is called the height of \alpha and denoted H(\alpha). If the leading coefficient of P is 1, \alpha is called algebraic integer. The degree of an algebraic integer \alpha is the degree of P. Also, let \gamma=\frac{1+\sqrt{5}}{2} denote the golden ratio.

The Theorem: On 11th October 2002, Damien Roy submitted to arxiv and Annals of Mathematics a paper in which he proved the existence of a transcendental real number \xi and constant c>0 such that, for any algebraic integer \alpha of degree at most 3, we have |\xi-\alpha| \geq c H(\alpha)^{-\gamma^2}.

Short context: For a positive integer n, let \tau_n be the supremum of all \tau \in {\mathbb R} such that for any transcendental \xi \in {\mathbb R} there exist infinitely many algebraic integers \alpha of degree at most n such that |\xi-\alpha| \leq H(\alpha)^{-\tau}. \tau_n measures the quality of approximation of real numbers by algebraic integers. It is known that \tau_2=2 and has been conjectured that \tau_n=n for all n\geq 2. In 1969, Davenport and Schmidt proved that \tau_3\geq \gamma^2. The Theorem implies that \tau_3\leq \gamma^2. Hence \tau_3=\gamma^2=\frac{3+\sqrt{5}}{2}<3, and the \tau_n=n conjecture is false.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 3.8 of this book for an accessible description of the Theorem.

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Published by Bogdan Grechuk

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