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A note on Diophantine approximation (II)

Published online by Cambridge University Press:  26 February 2010

H. Davenport
Affiliation:
Trinity College, Cambridge
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In 1956 Cassels proved the following result, which generalized a theorem of Marshall Hall on continued fractions. Let λ1 …, λr be any real numbers. Then there exists a real number α such that

for all integers u > 0 and for q = 1,…,r, where C = C(r) > 0. Thus all the numbers α+ λ1, …, α+ λr are badly approximable by rational numbers, which is equivalent to saying that the partial quotients in their continued fractions are bounded. In a previous paper I extended Cassels's result to simultaneous approximation. In the simplest case—that of simultaneous approximation to pairs of numbers—I proved that for any real λ1, …, λr and μ1, …, μr there exist α, β such that

for all integers u > 0 and for q=1,…, r, where again C = C(r) > 0. Both the construction of Cassels and my extension of it to more dimensions allow one to introduce an infinity of arbitrary choices, and consequently the set of α for (1) and the set of α, β for (2) may be made to have the cardinal of the continuum.

Type
Research Article
Copyright
Copyright © University College London 1964

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References

Cassels, J. W. S., “On a result of Marshall Hall”, Mathematika, 3 (1956), 109110.CrossRefGoogle Scholar

We write ‖θ‖ for the distance from θ to the nearest integer.

§ “A note on Diophantine approximation”, Studies in mathematical analysis and related topics, 7781, Stanford University Press, 1962.Google Scholar

Page52 note04 † The idea is that the transformation (9) approximates locally to a non-singular linear transformation, but I give the details for the sake of completeness.