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Limiting curlicue measures for theta sums

Francesco Cellarosi (2011)

Annales de l'I.H.P. Probabilités et statistiques

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...

Linear forms in the logarithms of three positive rational numbers

Curtis D. Bennett, Josef Blass, A. M. W. Glass, David B. Meronk, Ray P. Steiner (1997)

Journal de théorie des nombres de Bordeaux

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 Λ = b 2 log α 2 - b 1 log α 1 - b 3 log α 3 0 with b 1 , b 2 , b 3 positive integers and α 1 , α 2 , α 3 positive multiplicatively independent rational numbers greater than 1 . Let α j 1 = α j 1 / α j 2 with α j 1 , α j 2 coprime positive integers ( j = 1 , 2 , 3 ) . Let α j max { α j 1 , e } and assume that gcd ( b 1 , b 2 , b 3 ) = 1 . Let b ' = b 2 log α 1 + b 1 log α 2 b 2 log α 3 + b 3 log α 2 and assume that B max { 10 , log b ' } . We prove that either { b 1 , b 2 , b 3 } is c 4 , B -linearly dependent over (with respect to a 1 , a 2 , a 3 ) or Λ > exp - C B 2 j = 1 3 log a j , where c 4 and C = c 1 c 2 log ρ + δ are given in the tables...

Linear forms in two logarithms and interpolation determinants

Michel Laurent (1994)

Acta Arithmetica

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...

Linear forms of a given Diophantine type

Oleg N. German, Nikolay G. Moshchevitin (2010)

Journal de Théorie des Nombres de Bordeaux

We prove a result on the existence of linear forms of a given Diophantine type.

Linear fractional transformations of continued fractions with bounded partial quotients

J. C. Lagarias, J. O. Shallit (1997)

Journal de théorie des nombres de Bordeaux

Let θ be a real number with continued fraction expansion θ = a 0 , a 1 , a 2 , , and let M = a b c d be a matrix with integer entries and nonzero determinant. If θ has bounded partial quotients, then a θ + b c θ + d = a 0 * , a 1 * , a 2 * , also has bounded partial quotients. More precisely, if a j K for all sufficiently large j , then a j * | det ( M ) | ( K + 2 ) for all sufficiently large j . We also give a weaker bound valid for all a j * with j 1 . The proofs use the homogeneous Diophantine approximation constant L θ = lim sup q q q θ - 1 . We show that 1 det ( M ) L ( θ ) L a θ + b c θ + d det ( M ) L ( θ ) .

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