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Sequences of integers satisfying congruence relations and Pisot-Vijayaraghavan numbers

Published online by Cambridge University Press:  09 April 2009

Stephen Gerig
Affiliation:
The University of Western Australia
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We consider infinite sequences of positive integers having exponential growth: and becoming ultimately periodic modulo each member of a rather sparse infinite set of integers. If sufficient, natural conditions are placed on the growth and periodicities of , we find that a is an algebraic integer having all its algebraic conjugates within or on the unit circle, and fn has a special representation involving an. The result is a kind of dual to the theorem of Pisot (cf. Salem [2], p. 4, Theorem A).

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1967

References

[1]Cassels, J. W. S., An introduction to Diophantine approximation (Cambridge Tracts in Mathematics and Mathematical Physics, 45, Cambridge University Press, Cambridge, 1957).Google Scholar
[2]Salem, Raphael, Algebraic numbers and Fourier analysis (Heath, Boston, 1963).Google Scholar