The Conjugate Property of the Borel Theorem on Diophantine Approximation.
The continued fraction expansion of α with μ(α) = 3
The -dimensional Gauss transformation: Strong convergence and Lyapunov exponents.
The Farey Series of polynomials over a finite field.
The hyperbolic geometry of continued fractions .
The Jacobi-Perron Algorithm and Pisot numbers
The Lagrange theorem for multidimensional diophantine approximation.
The metrical theory of continued fractions to the nearer integer
The Modular Symbol and Continued Fractions in Higher Dimensions.
The Number of Steps in a Finite Jacobi Algorithm.
The √p Riemann surface
The Quality of the Diophantine Approximations Found by the Jacobi-Perron Algorithm and Related Algorithms.
The sequence of fractional parts of roots
We study the function , where θ is a positive real number, ⌊·⌋ and · are the floor and fractional part functions, respectively. Nathanson proved, among other properties of , that if log θ is rational, then for all but finitely many positive integers n, . We extend this by showing that, without any condition on θ, all but a zero-density set of integers n satisfy . Using a metric result of Schmidt, we show that almost all θ have asymptotically (log θ log x)/12 exceptional n ≤ x. Using continued...
The three-dimensional Gauss algorithm is strongly convergent almost everywhere.
Tong’s spectrum for Rosen continued fractions
In the 1990s, J.C. Tong gave a sharp upper bound on the minimum of 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 approximation...
Trajectories of rotations
Transcendence with Rosen continued fractions
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.
Transcendental numbers having explicit -adic and Jacobi-Perron expansions
Two theorems on meromorphic functions used as a principle for proofs on irrationality
In this paper we discuss two theorems on meromorphic functions of Nikishin and Chudnovsky. Our purpose is to show, how to derive some well-known but not obvious results on irrationality in a systematic and simple way from properties of meromorphic functions with arithmetic conditions. As far as it stands, we have no new results on irrationality, to the contrary some results on numbers of the corollaries are known already since a long time to be transcendental (cf. [4], [9] and [10]). Our main intention...