Learn More
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)
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)
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)
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)
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)