Damien Roy

Approximation to real numbers by cubic algebraic integers I

In 1969, H. Davenport and W. M. Schmidt studied the problem of approximation to a real number

Approximation simultanée d'un nombre et de son carré

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Criteria of algebraic independence with multiplicities and interpolation determinants

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

On Two Exponents of Approximation Related to a Real Number and Its Square

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On Schmidt and Summerer parametric geometry of numbers

and

Simultaneous Approximation and Algebraic Independence

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Diophantine approximation by conjugate algebraic integers

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

A note on Siegel's Lemma over number fields

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On the Continued Fraction Expansion of a Class of Numbers

. 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.

On simultaneous rational approximations to a real number, its square, and its cube

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 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).