A new multidimensional continued fraction algorithm

Abstract

Abstract. It has been believed that the continued fraction expansion of (α, β) (1, α, β is a Q-basis of a real cubic field) obtained by the modified JacobiPerron algorithm is periodic. We conducted a numerical experiment (cf. Table B, Figure 1 and Figure 2) from which we conjecture the non-periodicity of the expansion of () ( x denoting the fractional part of x). We present a new algorithm which is something like the modified Jacobi-Perron algorithm, and give some experimental results with this new algorithm. From our experiments, we can expect that the expansion of (α, β) with our algorithm always becomes periodic for any real cubic field. We also consider real quartic fields. IntroductionThe study of continued fractions has a long history dating back to J. Wallis (1616-1703) and Ch. Huygens (1629-1695) [8]. In particular, many kinds of higherdimensional continued fractions have been studied starting with K.G. Jacobi (1804-1851) [7]. A central problem has been to find a higher-dimensional generalization of Legendre's theorem concerning the periodic continued fractions. In fact, the following conjecture has been believed. 1, α 1 Figure 1 and Figure 2) from which we conjecture the non-periodicity of the expansion of ( Conjecture. Let) by the modified Jacobi-Perron algorithm.In this paper, we give some candidates of algorithms of continued fraction expansion of dimensions 2 and 3, which can be easily generalized to any dimension.By numerical experiments, we checked that, for instance, ( √ m / ∈ Q) obtained by our algorithm become periodic. We showed the periodicity

True orbit simulation of piecewise linear and linear fractional maps of arbitrary dimension using algebraic numbers

“…On the other hand, Tamura and Yasutomi made the opposite conjecture that the expansion of ð ffiffi ffi 3 3 p À1; ffiffi ffi 3 3 p 2 À 2Þ by the MJPA is not eventually periodic. 27 We estimated the marginal densities using the true orbit starting from ð ffiffi ffi 3 3 p À1; ffiffi ffi 3 3 p 2 À 2Þ, and we obtained similar results as shown in Fig. 5 saying that, in this case as well, the estimated marginal densities agree with those of the map associated with the MJPA.…”

“…25 Also, true orbits of piecewise linear fractional maps are often generated for studying continued fractions. [26][27][28] From the viewpoint of simulation, however, we must pay further attention to the following point. In simulations, it is also necessary to reproduce behaviors typically exhibited by a target system, but true orbits alone do not necessarily mean they are typical.…”

On $p$-adic multidimensional continued fractions

“…In particular, the theorem provides some conditions for constructing several functions s (i) useful for constructing iterative algorithms. All the functions appearing in literature (as the function used by Browkin [6], Ruban [29], Schneider [32], Tamura and Yasutomi [34]) can be seen as particular realizations of the functions s (i) described below.…”

Numeration and discrete dynamical systems

“…It was introduced by Jacobi, and then later by Perron, in order to characterize cubic numbers as numbers having periodic expansions. Numerical evidence does not support this belief anymore (see [93] for an algorithm aiming at characterizing cubic numbers). Nevertheless, Jacobi-Perron algorithm and Ostrowski algorithms behave in completely different ways, arithmetically or as dynamical systems.…”