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- Damien Roy
- 2009

Abstract. 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 by rational numbers with the same denominator. In this paper, we show that their… (More)

- 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), where H(α) denotes the height of α. Although this is true for n = 2, we show here that, for n = 3, the optimal exponent… (More)

- Damien Roy
- 2008

The study of approximation to a real number by algebraic numbers of bounded degree started with a paper of E. Wirsing [10] in 1961. Motivated by this, H. Davenport and W. M. Schmidt considered in [5] the analogous inhomogeneous problem of approximation to a real number by algebraic integers of bounded degree. They proved a result that is optimal for degree… (More)

- Benoit Arbour, Damien Roy
- 2008

then ξ is algebraic over Q of degree at most n. For example, Brownawell’s version of Gel’fond’s criterion in [1] implies that the above statement holds with any τ > 3n, and the more specific version proved by Davenport and Schmidt as Theorem 2b of [4] shows that it holds with τ = 2n−1. On the other hand, the above application of Dirichlet box principle… (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 polynomials of bounded degree taking small values at ξ together with most of their derivatives. The… (More)

- DAMIEN ROY
- 1997

We establish several new measures of simultaneous algebraic approximations for families of complex numbers (θ1, . . . , θn) 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 fieldQ(θ1, . . . , θn) has… (More)

- DAMIEN ROY
- 2008

We show that, for any transcendental real number ξ, the uniform exponent of simultaneous approximation of the triple (ξ, ξ, ξ) by rational numbers with the same denominator is at most (1 + 2γ − √ 1 + 4γ)/2 ∼= 0.4245 where γ = (1 + √ 5)/2 stands for the golden ratio. As a consequence, we get a lower bound on the exponent of approximation of such a number ξ… (More)

- DAMIEN ROY
- 2003

Here, the exponent of q in the upper bound is optimal because, when ξ has bounded partial quotients, there is also a constant c > 0 such that |ξ − p/q| ≥ cq for all rational numbers p/q (see Chapter I of [14]). Define the height H(P ) of a polynomial P ∈ R[T ] as the largest absolute value of its coefficients, and the height H(α) of an algebraic number α as… (More)

- DAMIEN ROY
- 2008

For each real number ξ, let λ̂2(ξ) denote the supremum of all real numbers λ such that, for each sufficiently large X , the inequalities |x0| ≤ X , |x0ξ − x1| ≤ X and |x0ξ − x2| ≤ X admit a solution in integers x0, x1 and x2 not all zero, and let ω̂2(ξ) denote the supremum of all real numbers ω such that, for each sufficiently large X , the dual… (More)

- DAMIEN ROY
- 2004

We present a general result of simultaneous approximation to several transcendental real, complex or p-adic numbers ξ1, ..., ξt by conjugate algebraic numbers of bounded degree over Q, provided that the given transcendental numbers ξ1, ..., ξt generate over Q a field of transcendence degree one. We provide sharper estimates for example when ξ1, ..., ξt form… (More)