Multidimensional Continued Fractions

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 three aspects. Invariant measures: For several algorithms the invariant measures are known. However, for Jacobi-Perron algorithm up to now no explicit formula has been found. The application of dual algorithms lead to integration over measures concentrated on fractal like sets which needs further investigation. 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. For 2-dimensional algorithms several results are already known. But for dimension 3 the situation is worse. 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 method 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. It would be an important question to generalise this method to multidimensional continued fractions.

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. Since the days of Jacobi for the multidimensional case several algorithms have been proposed. In the named project new results for dimension two have been obtained. The quality of approximation for Selmer`s algorithm and the algorithm of Selmer-Greiter was determined. Furthermore the existence of an interesting set which may remind to fractal sets was shown for a subtractive algorithm. The method of singularisation which connects several variants of continued fractions was extended for the first time to two-dimensional algorithms.