The Szekeres multidimensional continued fraction
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 .
-  A. Baker, A sharpening of the bounds for linear forms in logarithms, Acta Arith. 21 (1972), 117–129. MR 0302573, https://doi.org/10.4064/aa-21-1-117-129
-  T. W. Cusick, Diophantine approximation of ternary linear forms, Math. Comp. 25 (1971), 163–180. MR 0296022, https://doi.org/10.1090/S0025-5718-1971-0296022-4
-  T. W. Cusick, Diophantine approximation of ternary linear forms. II, Math. Comp. 26 (1972), 977–993. MR 0321879, https://doi.org/10.1090/S0025-5718-1972-0321879-9
-  T. W. Cusick, Diophantine approximation of linear forms over an algebraic number field, Mathematika 20 (1973), 16–23. MR 0340186, https://doi.org/10.1112/S0025579300003582
-  J. F. Koksma, Diophantische Approximationen, Springer-Verlag, Berlin-New York, 1974 (German). Reprint. MR 0344200
-  Serge Lang, Introduction to diophantine approximations, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1966. MR 0209227
-  William J. LeVeque (ed.), Reviews in number theory. Vol. 1, American Mathematical Society, Providence, R.I., 1974. Reviews reprinted from Mathematical Reviews, Vols. 1–44, published during 1940–1972. MR 0349547
-  H. MINKOWSKI, "Zur Theorie der Kettenbrüche," in Gesammelte Abhandlungen, Vol. I, Teubner, Leipzig, 1911, pp. 278-292.
-  G. Szekeres, Multidimensional continued fractions, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 13 (1970), 113–140 (1971). MR 0313198
- 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)
- T. W. CUSICK, "Diophantine approximation of ternary linear forms," Math. Comp., v. 25, 1971, pp. 163-180. MR 45 #5083. MR 0296022 (45:5083)
- 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)
- 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)
- J. F. KOKSMA, Diophantische Approximationen, reprint of 1936 Springer edition, Chelsea, New York. MR 0344200 (49:8940)
- S. LANG, Introduction to Diophantine Approximations, Addison-Wesley, Reading, Mass., 1966. MR 35 #129. MR 0209227 (35:129)
- W. J. LeVEQUE (Editor), Reviews in Number Theory, Vol. 1, Amer. Math. Soc., Providence, R. I., 1974. MR 50 #2040. MR 0349547 (50:2040)
- H. MINKOWSKI, "Zur Theorie der Kettenbrüche," in Gesammelte Abhandlungen, Vol. I, Teubner, Leipzig, 1911, pp. 278-292.
- 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
Keywords: Szekeres multidimensional continued fraction, ternary linear forms, Diophantine inequality, totally real cubic field
Article copyright: © Copyright 1977 American Mathematical Society