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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 ) .

Markoff numbers and ambiguous classes

Anitha Srinivasan (2009)

Journal de Théorie des Nombres de Bordeaux

The Markoff conjecture states that given a positive integer c , there is at most one triple ( a , b , c ) of positive integers with a b c that satisfies the equation a 2 + b 2 + c 2 = 3 a b c . The conjecture is known to be true when c is a prime power or two times a prime power. We present an elementary proof of this result. We also show that if in the class group of forms of discriminant d = 9 c 2 - 4 , every ambiguous form in the principal genus corresponds to a divisor of 3 c - 2 , then the conjecture is true. As a result, we obtain criteria in terms of...

On the closedness of approximation spectra

Jouni Parkkonen, Frédéric Paulin (2009)

Journal de Théorie des Nombres de Bordeaux

Generalizing Cusick’s theorem on the closedness of the classical Lagrange spectrum for the approximation of real numbers by rational ones, we prove that various approximation spectra are closed, using penetration properties of the geodesic flow in cusp neighbourhoods in negatively curved manifolds and a result of Maucourant [Mau].

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