Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Irrationality of limits of quickly convergent algebraic numbers sequences


Author: A. V. Nabutovsky
Journal: Proc. Amer. Math. Soc. 102 (1988), 473-479
MSC: Primary 11J72; Secondary 40A05
DOI: https://doi.org/10.1090/S0002-9939-1988-0928963-0
MathSciNet review: 928963
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We present criteria for irrationality of limits of convergent sequences of rational numbers, algebraic numbers of the same degree and of strictly increasing degrees.

The criterion for irrationality of limits of a sequence of rational numbers has the form of an infinite system of inequalities on successive differences between elements. These inequalities are not strict. If these inequalities are satisfied, then the limit is rational if and only if all inequalities but a finite set of them are satisfied as equalities and the sequence becomes monotonous beginning from some element. So, the criterion permits to see a "border" between rationality and irrationality for some class of quickly convergent sequences.


References [Enhancements On Off] (What's this?)

  • [1] P. Erdo Problem 4321, Amer. Math. Monthly 64 (1957), 47.
  • [2] Maurice Mignotte, Approximation des nombres algébriques par des nombres algébriques de grand degré, Ann. Fac. Sci. Toulouse Math. (5) 1 (1979), no. 2, 165–170 (French, with English summary). MR 554376
  • [3] Wolfgang M. Schmidt, Diophantine approximation, Lecture Notes in Mathematics, vol. 785, Springer, Berlin, 1980. MR 568710
  • [4] D. Perron, Irrationalzahlen, DeGruyter, Berlin, 1960.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 11J72, 40A05

Retrieve articles in all journals with MSC: 11J72, 40A05


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0928963-0
Article copyright: © Copyright 1988 American Mathematical Society