Speaker: Anthony Poels and Damien Roy (University of Ottawa)
8 October, 2021 14:00 UTC
Abstract: Let $\xi\in\mathbb{R}\setminus\bar{\mathbb{Q}}$ be a real transcendental number and let $n$ be a positive integer. By pioneer work of Davenport and Schmidt from 1969, we know that the exponent $\tau_{n+1}(\xi)$ of best approximation to $\xi$ by algebraic integers of degree at most $n+1$ is at least equal to $1+1/\lambda_n(\xi)$, where $\lambda_n(\xi)$ stands for the uniform exponent of rational approximation to the successive powers $1,\xi,\dots,\xi^n$ of $\xi$. So any upper bound on $\lambda_n(\xi)$ which holds for any $\xi\in\mathbb{R}\setminus\bar{\mathbb{Q}}$ provides a lower bound on $\tau_{n+1}(\xi)$ which is also independent of $\xi$. In this talk, we present new tools which yield, for each integer $n\ge 4$, a significantly improved upper bound on $\lambda_n(\xi)$ and thus a refined lower bound on $\tau_{n+1}(\xi)$. The new lower bound is $n/2+a\sqrt{n}+4/3$ with $a=(1-\log(2))/2\simeq 0.153$, instead of the current $n/2+\mathcal{O}(1)$.
As usual, the starting point is the sequence of so-called minimal points $x_1,x_2,x_3,\dots$ in $\mathbb{Z}^{n+1}$ defined initially by Davenport and Schmidt. Our strategy consists in estimating from above the height of the subspaces of $\mathbb{R}^{n-\ell+1}$ generated by $n-\ell+1$ consecutive coordinates from each point among $x_i,x_{i+1},\dots,x_q$ for given $i\le q$. To this end, we first need a lower bound for the dimension of such spaces.
In the first part of the talk, we present the required background with some historical perspective and our key algebraic result concerning the dimension of the above mentioned spaces. In the second part, we look at their heights from different perspectives and outline the general strategy of the proof.