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

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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.

**[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.

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DOI:
https://doi.org/10.1090/S0002-9939-1988-0928963-0

Article copyright:
© Copyright 1988
American Mathematical Society