Diophantine Properties of Multidimensional C.F.

C. G. J. Jacobi (1804-1851) was the first mathematician who studied multidimensional continued fractions. His main objective was to find a generalisation of Lagrange`s famous theorem which characterises quadratic irrational numbers as periodic continued fractions. He proposed an algorithm for number pairs in the hope that this algorithm will become eventually periodic if and only if both numbers belong to a cubic number field. Jacobi found some examples, but he could not show that Lagrange`s theorem is true for this algorithm. In fact, the problem is still open! Following Jacobi other multidimensional continued fraction algorithms were proposed by several authors. During the last years multidimensional continued fractions became interesting also for applications in physics (renormalisation theory, percolation theory) and numerical mathematics (addition chains). A survey of both algebraic and ergodic aspects can be found in Schweiger`s recent book Multidimensional Continued Fractions. The proposed research centres around two related aspects. Diophantine properties: Multidimensional continued fractions produce infinitely many points with rational coordinates. The problem is to estimate the quality of the approximation. Lagarias was able to connect this problem with the Lyapunov exponents of the underlying ergodic chain. In recent papers Schweiger extended these results to Baldwin`s algorithm and Greiter`s modification of Selmer`s algorithm. In a long paper Broise-Alamichel & Guivarc`h successfully applied smooth ergodic theory to obtain new insights in the Lyapunov spectrum. An explicit construction of the eigenvectors is possible for dimension 2 (Schweiger, unpublished). However, for dimension 3 and higher dimensions the situation is still unsatisfactory. Schratzberger was able to prove a result for the multiplicative version of Brun`s algorithm. It would be important to investigate other algorithms and hopefully to simplify the methods to approach higher dimensions. Singularisation: For 1-dimensional continued fractions Kraaikamp has shown in a series of papers that the method of singularisation can be used to understand the connections between several types of 1-dimensional continued fractions. In a recent paper (submitted for publication) Schratzberger has applied this method to construct a "quadratic acceleration" of 2-dimensional Brun`s algorithm. In his subsequent investigations Schratzberger could show that in many cases Brun`s algorithm and Jacobi algorithm can be converted which strongly suggests that their approximation qualities are equal. It would be very important to complete these investigations and to extend to other algorithms.

In number theory the question of approximation of real numbers by rational numbers is of great importance. This area is called Diophantine Approximation, in honour of the Greek mathematician Diophantos of Alexandria. In the one-dimensional case (approximation of a single real number by rational numbers) the powerful theory of continued fractions and its variants is available. In dimension two the algorithm of Jacobi-Perron and Brun`s algorithm are most widely studied. The method of singularisation which connects several variants of continued fractions was extended for the first time to two-dimensional algorithms. The main goal was to show that both algorithms are related in the following sense. With the possible exception of a set of measure zero one algorithm can be converted into the other. The method mixes theoretical considerations with a heavy amount of computer assisted calculations. An important side result was the extension of Lévy`s method to Brun`s algorithm.