Limit points in the Lagrange spectrum of a quadratic field
We consider the ensemble of curves {γα, N: α∈(0, 1], N∈ℕ} obtained by linearly interpolating the values of the normalized theta sum N−1/2∑n=0N'−1exp(πin2α), 0≤N'<N. We prove the existence of limiting finite-dimensional distributions for such curves as N→∞, when α is distributed according to any probability measure λ, absolutely continuous w.r.t. the Lebesgue measure on [0, 1]. Our Main Theorem generalizes a result by Marklof [Duke Math. J.97 (1999) 127–153] and Jurkat and van Horne [Duke...
In this paper we prove a lower bound for the linear dependence of three positive rational numbers under certain weak linear independence conditions on the coefficients of the linear forms. Let with positive integers and positive multiplicatively independent rational numbers greater than . Let with coprime positive integers . Let and assume that gcd Letand assume that We prove that either is -linearly dependent over (with respect to ) orwhere and are given in the tables...
1. Introduction. Our aim is to test numerically the new method of interpolation determinants (cf. [2], [6]) in the context of linear forms in two logarithms. In the recent years, M. Mignotte and M. Waldschmidt have used Schneider's construction in a series of papers [3]-[5] to get lower bounds for such a linear form with rational integer coefficients. They got relatively precise results with a numerical constant around a few hundreds. Here we take up Schneider's method again in the framework...
We prove a result on the existence of linear forms of a given Diophantine type.
Let be a real number with continued fraction expansionand letbe a matrix with integer entries and nonzero determinant. If has bounded partial quotients, then also has bounded partial quotients. More precisely, if for all sufficiently large , then for all sufficiently large . We also give a weaker bound valid for all with . The proofs use the homogeneous Diophantine approximation constant . We show that