Displaying similar documents to “A note on counting cuspidal excursions.”

Transcendence with Rosen continued fractions

Yann Bugeaud, Pascal Hubert, Thomas A. Schmidt (2013)

Journal of the European Mathematical Society

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We give the first transcendence results for the Rosen continued fractions. Introduced over half a century ago, these fractions expand real numbers in terms of certain algebraic numbers.

Continued fraction expansions for complex numbers-a general approach

S. G. Dani (2015)

Acta Arithmetica

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We introduce a general framework for studying continued fraction expansions for complex numbers, and establish some results on the convergence of the corresponding sequence of convergents. For continued fraction expansions with partial quotients in a discrete subring of ℂ an analogue of the classical Lagrange theorem, characterising quadratic surds as numbers with eventually periodic continued fraction expansions, is proved. Monotonicity and exponential growth are established for the...

Tong’s spectrum for Rosen continued fractions

Cornelis Kraaikamp, Thomas A. Schmidt, Ionica Smeets (2007)

Journal de Théorie des Nombres de Bordeaux

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In the 1990s, J.C. Tong gave a sharp upper bound on the minimum of k consecutive approximation constants for the nearest integer continued fractions. We generalize this to the case of approximation by Rosen continued fraction expansions. The Rosen fractions are an infinite set of continued fraction algorithms, each giving expansions of real numbers in terms of certain algebraic integers. For each, we give a best possible upper bound for the minimum in appropriate consecutive blocks of...

Continued fractions on the Heisenberg group

Anton Lukyanenko, Joseph Vandehey (2015)

Acta Arithmetica

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We provide a generalization of continued fractions to the Heisenberg group. We prove an explicit estimate on the rate of convergence of the infinite continued fraction and several surprising analogs of classical formulas about continued fractions.