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On the fractional parts of the powers of a number (II)

Published online by Cambridge University Press:  24 October 2008

T. Vijayaraghavan
T. Vijayaraghavan
Affiliation:
Department of MathematicsDacca University, India
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Extract

Let G(θ) denote the limit points of the fractional parts of θn (n = 1, 2, 3, …), and T the set of numbers θ for which G(θ) contains an infinity of points. In an earlier note it was proved that if θ is a rational fraction greater than unity then θ belongs to T. The results of this note concern the limit points of the powers of algebraic irrationals.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 37 , Issue 4 , October 1941 , pp. 349 - 357
Copyright
Copyright © Cambridge Philosophical Society 1941

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References

* J. London Math. Soc. 15 (1940), 159–60.Google Scholar

† It is not supposed that k ≥ 3; indeed k may be equal to zero.

* Suppose on the contrary that α is a rational multiple of π. Let p be such that pα is an integral multiple of 2π. Now the polynomial has integer coefficients, and has two equal roots, on account of the choice of p. Hence is an algebraic integer of degree 1 or 2. But it is not of degree 1 since and it is not of degree 2 either, since the constant term in is equal to which is not a rational number. Hence α is an irrational multiple of π.

* A necessary and sufficient condition to ensure the irreducibility of P(x) is that

{a 2 + (b + m 2 + 2)2} (b + 2 + 2a) (b + 2 − 2a) (a 2 − 4b + 8 − m 2) ≠ 0

for m = 0,1,2,3, …; a sufficient condition to secure that the equation P(x) = 0 has two real positive roots and two non-real roots is that 0 > b ≥ a.

† If α were a rational multiple of π, then P(x) would be divisible by a cyclotomic polynomial, which is not possible since P(x) is irreducible and has zeros which do not lie on the unit circle.

* Acta Math. 30 (1906), 369.Google Scholar

† Math. Annalen 77 (1916), 510–12Google Scholar. See also Pólya, G. and Szegö, G., Aufgaben und Lehrsätze aus der Analysis, 2 (1925), 142 and 357.Google Scholar

‡ Hardy, G. H., J. Indian Math. Soc. 11 (1919), 162–166Google Scholar, gives the special case of this theorem in which the sequence is assumed to have one limit point only.

* As soon as it was proved that b 1b 2 … b l ≠ 0 we could have deduced from the theorem mentioned below that ξ1, ξ2, …, ξ1 are algebraic integers. Let g 1(z), g 2(z), …, g m(z) be non-vanishing integral functions of z, and let M ν(z) denote the maximum modulus of g ν(z) on the circle | z | = r. If r −1 log M ν(r) → 0 as r → ∞ (ν = 1, 2, …, m), and

has rational integral values for z = 0, 1, 2, 3, …, then g 1(z), g 2(z), …, g m(z) are polynomials and ξ1, ξ2, …, ξmare algebraic integers [Pólya, G., Math. Annalen, 77 (1916), 512CrossRefGoogle Scholar]. In connexion with this theorem, it may be stated that it is also true that, if a number η is a conjugate of one or more of the numbers ξ1, ξ2, …, ξm, then η is one of the numbers ξ1, ξ2, …, ξm.

* Since

* Pólya, G. and Szegö, G., Aufgaben, 2 (1925), 149 and 368.Google Scholar