The Szekeres multidimensional continued fraction

Author:
T. W. Cusick

Journal:
Math. Comp. **31** (1977), 280-317

MSC:
Primary 10F20

DOI:
https://doi.org/10.1090/S0025-5718-1977-0429765-5

MathSciNet review:
0429765

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In his paper "Multidimensional continued fractions" (*Ann. Univ. Sci. Budapest. Eötvös Sect. Math.*, v. 13, 1970, pp. 113-140), 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 *k*-tuple of real numbers (satisfying ) a sequence of positive integers; this sequence is called a continued *k*-fraction, 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 2-fraction 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 2-fraction of are proved. The method used can be applied to other pairs of cubic irrationals .

**[1]**ALAN BAKER, "A sharpening of the bounds for linear forms in logarithms,"*Acta Arith.*, v. 21, 1972, pp. 117-129. MR**46**#1717. MR**0302573 (46:1717)****[2]**T. W. CUSICK, "Diophantine approximation of ternary linear forms,"*Math. Comp.*, v. 25, 1971, pp. 163-180. MR**45**#5083. MR**0296022 (45:5083)****[3]**T. W. CUSICK, "Diophantine approximation of ternary linear forms. II,"*Math. Comp.*, v. 26, 1972, pp. 977-993. MR**48**#244. MR**0321879 (48:244)****[4]**T. W. CUSICK, "Diophantine approximation of linear forms over an algebraic number field,"*Mathematika*, v. 20, 1973, pp. 16-23. MR**49**#4942. MR**0340186 (49:4942)****[5]**J. F. KOKSMA,*Diophantische Approximationen*, reprint of 1936 Springer edition, Chelsea, New York. MR**0344200 (49:8940)****[6]**S. LANG,*Introduction to Diophantine Approximations*, Addison-Wesley, Reading, Mass., 1966. MR**35**#129. MR**0209227 (35:129)****[7]**W. J. LeVEQUE (Editor),*Reviews in Number Theory*, Vol. 1, Amer. Math. Soc., Providence, R. I., 1974. MR**50**#2040. MR**0349547 (50:2040)****[8]**H. MINKOWSKI, "Zur Theorie der Kettenbrüche," in*Gesammelte Abhandlungen*, Vol. I, Teubner, Leipzig, 1911, pp. 278-292.**[9]**G. SZEKERES, "Multidimensional continued fractions,"*Ann. Univ. Sci. Budapest. Eötvös Sect. Math.*, v. 13, 1970, pp. 113-140. MR**47**#1753. MR**0313198 (47:1753)**

Retrieve articles in *Mathematics of Computation*
with MSC:
10F20

Retrieve articles in all journals with MSC: 10F20

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1977-0429765-5

Keywords:
Szekeres multidimensional continued fraction,
ternary linear forms,
Diophantine inequality,
totally real cubic field

Article copyright:
© Copyright 1977
American Mathematical Society