Damien Roy

In 1969, H. Davenport and W. M. Schmidt studied the problem of approximation to a real number $\xi$ by algebraic integers of degree at most 3. They did so, using geometry of numbers, by resorting to the dual problem of finding simultaneous approximations to $\xi$ and $\xi^2$ by rational numbers with the same denominator. In this paper, we show that their measure of approximation for the dual problem is optimal and that it is realized for a countable set of real numbers $\xi$. We give several properties of these numbers including measures of approximation by rational numbers, by quadratic real numbers and by algebraic integers of degree at most 3.

2000 Mathematical Subject Classification: 11J04 (primary), 11J13, 11J82 (secondary).

Keywords: simultaneous approximation, algebraic numbers, algebraic integers, measure of approximation, continued fractions.

E-mail: droy@uottawa.ca