Approximation simultanée d'un nombre et de son carré

In 1969, H. Davenport and W.M. Schmidt established a measure of the simultaneous approximation for a real number ξ and its square by rational numbers with the same denominator, assuming only that ξ is not rational nor quadratic over $\text{Q}$. Here, we show by an example, that this measure is optimal. We also indicate several properties of the numbers for which this measure is optimal, in particular with respect to approximation by algebraic integers of degree at most three. To cite this article: D. Roy, C. R. Acad. Sci. Paris, Ser. I 336 (2003).