The Szekeres multidimensional continued fraction
Author:
T. W. Cusick
Journal:
Math. Comp. 31 (1977), 280317
MSC:
Primary 10F20
MathSciNet review:
0429765
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Abstract: In his paper "Multidimensional continued fractions" (Ann. Univ. Sci. Budapest. Eötvös Sect. Math., v. 13, 1970, pp. 113140), G. Szekeres introduced a new higher dimensional analogue of the ordinary continued fraction expansion of a single real number. The Szekeres algorithm associates with each ktuple of real numbers (satisfying ) a sequence of positive integers; this sequence is called a continued kfraction, and for k = 1 it is just the sequence of partial quotients of the ordinary continued fraction for . A simple recursive procedure applied to produces a sequence and are integers) of simultaneous rational approximations to and a sequence of integer tuples such that the linear combination approximates zero. Szekeres conjectured, on the basis of extensive computations, that the sequence contains all of the "best" simultaneous rational approximations to and that the sequence contains all of the "best" approximations to zero by the linear form . For the special case k = 2 and (where is the positive root of , Szekeres further conjectured that the 2fraction is "almost periodic" in a precisely defined sense. In this paper the Szekeres conjectures concerning best approximations to zero by the linear form and concerning almost periodicity for the 2fraction of are proved. The method used can be applied to other pairs of cubic irrationals .
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197704297655
PII:
S 00255718(1977)04297655
Keywords:
Szekeres multidimensional continued fraction,
ternary linear forms,
Diophantine inequality,
totally real cubic field
Article copyright:
© Copyright 1977
American Mathematical Society
