An analogue of the nearest integer continued fraction for certain cubic irrationalities

Abstract

Abstract. Let 8 be any irrational and define Ne(9) to be that integer such that \8 -Ne{6)\ < \. Put p0 = 6, r0 = Ne(p0), pk+ | = \/(rk -pk), rk+x = Ne(pk+I). Then the r's here are the partial quotients of the nearest integer continued fraction (NICF) expansion of 9. When D is a positive nonsquare integer, and 6 = {D, this expansion is periodic. It can be used to find the regulator of 2(/JD ) in less than 75 percent of the time needed by the usual continued fraction algorithm. A geometric interpretation of this algorithm is given and this is used to extend the NICF to a nearest integer analogue of the Voronoi Continued Fraction, which is used to find the regulator of a cubic field íwith negative discriminant A. This new algorithm (NIVCF) is periodic and can be used to find the regulator of §.If / < ^/|A j/148, the NIVCF algorithm can be used to find any algebraic integer a of € such that N(a) = I. Numerical results suggest that the NIVCF algorithm finds the regulator of f = 2(/ö ) in about 80 percent of the time needed by Voronoi's algorithm.

On the metric theory of nearest integer continued fractions

“…We shall use the term nearest iiueger continued fraction for such a representation. (Please, observe that this definition does not coincide with the one given in [5].) One of the advantages of the representation we are choosing for the nearest integer continued fraction is that the remainder after n denominators, wn, is always in the interval (-½, ½).…”

The spacing of the minima in certain cubic lattices

“…This is a semi-open right circular cylinder, symmetric about the origin O of <f 3 , with axis the x-axis of S y It should be mentioned at this point that if α,β<ΞJT and |α'| = \β% then a = ±β (see [1], p. 274). Thus, if |β'| = |α'|, then B £ Jir(a).…”

“…The methods used to prove these results are completely elementary. We shall need to make use of these results together with several others established in [3]; however, we first give some simple lemmas concerning points of St. Throughout this work we will use θ to denote the minimum of 3t adjacent to 1, ω to denote the minimum of 9t adjacent to 0, and χ to denote the minimum of 31 adjacent to ω.…”

“…We have |γ'| < 1; thus, if κ(γ) > 1, then |1 -a' -γ'| = \β'\ > 1, which is not so. D We will also require some lemmas whose proofs have already appeared in [3]. We will only give the statements of these results here; however, we mention that the proofs of these lemmas are elementary and require, for the most part, only results from simple plane geometry.…”

On the periodic writing of cubic irrationals and a generalization of Rédei functions

“…Moreover, Tamura and Yasutomi [33], [34] recently presented a modified Jacobi-Perron algorithm. Similarly, the Jacobi-Perron algorithm has been modified using different functions, e.g., in [17], [37], [25], [32]. See the beautiful book of Schweiger [31] for a guide about multidimensional continued fractions and [21] for a new geometric vision of multidimensional continued fractions.…”