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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)
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)
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)
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)
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)
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)
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)