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- Damien Roy
- 2009

In 1969, H. Davenport and W. Schmidt studied the problem of approximation to a real number ξ by algebraic integers of degree at most three. They did so, using geometry of numbers, by resorting to the dual problem of finding simultaneous approximations to ξ and ξ 2 by rational numbers with the same denominator. In this paper, we show that their measure of… (More)

- Damien Roy
- 2008

- Damien Roy, DAMIEN ROY
- 2004

It has been conjectured for some time that, for any integer n ≥ 2, any real number ǫ > 0 and any transcendental real number ξ, there would exist infinitely many algebraic integers α of degree at most n with the property that |ξ − α| ≤ H(α) −n+ǫ , where H(α) denotes the height of α. Although this is true for n = 2, we show here that, for n = 3, the optimal… (More)

Building on the work of Davenport and Schmidt, we mainly prove two results. The first one is a version of Gel'fond's transcendence criterion which provides a sufficient condition for a complex or p-adic number ξ to be algebraic in terms of the existence of polynomi-als of bounded degree taking small values at ξ together with most of their derivatives. The… (More)

- DAMIEN ROY
- 2008

For each real number ξ, letˆλ 2 (ξ) denote the supremum of all real numbers λ such that, for each sufficiently large X, the inequalities |x 0 | ≤ X, |x 0 ξ − x 1 | ≤ X −λ and |x 0 ξ 2 − x 2 | ≤ X −λ admit a solution in integers x 0 , x 1 and x 2 not all zero, and letˆω 2 (ξ) denote the supremum of all real numbers ω such that, for each sufficiently large X,… (More)

- DAMIEN ROY
- 2003

- DAMIEN ROY
- 1997

We establish several new measures of simultaneous algebraic approximations for families of complex numbers (θ 1 ,. .. , θ n) related to the classical exponential and elliptic functions. These measures are completely explicit in terms of the degree and height of the algebraic approximations. In some instances, they imply that the field Q(θ 1 ,. .. , θ n) has… (More)

- Dmitrij Zelo, Damien Roy
- 2008

In this thesis, we study the problem of simultaneous approximation to a fixed family of real and p-adic numbers by roots of integer polynomials of restricted type. The method that we use for this purpose was developed by H. Davenport and W.M. Schmidt in their study of approximation to real numbers by algebraic integers. This method based on Mahler's Duality… (More)

- Benoit Arbour, Damien Roy
- 2008

- DAMIEN ROY
- 2004

We present a general result of simultaneous approximation to several transcen-dental real, complex or p-adic numbers ξ 1 , ..., ξ t by conjugate algebraic numbers of bounded degree over Q, provided that the given transcendental numbers ξ 1 , ..., ξ t generate over Q a field of transcendence degree one. We provide sharper estimates for example when ξ 1 ,… (More)