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An Inhomogeneous Minimum For Non-Convex Star-Regions With Hexagonal Symmetry

Published online by Cambridge University Press:  20 November 2018

R. P. Bambah  and
K. Rogers
R. P. Bambah
Affiliation:
Punjab University
K. Rogers
Affiliation:
Princeton University
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1. Introduction. Several authors have proved theorems of the following type:

Let x0, y0 be any real numbers. Then for certain functions f(x, y), there exist numbers x, y such that

1.1 x ≡ x0, y ≡ y0 (mod 1),

and

1.2 .

The first result of this type, but with replaced by min , was given by Barnes (3) for the case when the function is an indefinite binary quadratic form. A generalisation of this was proved by elementary geometry by K. Rogers (6).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 7 , 1955 , pp. 337 - 346
Copyright
Copyright © Canadian Mathematical Society 1955

References

1. Bambah, R. P., Non-homogeneous binary cubic forms, Proc. Camb. Phil. Soc., 47 (1951), 457–460.Google Scholar
2. Bambah, R. P., On the geometry of numbers of non-convex star-regions with hexagonal symmetry, Phil. Trans. Royal Soc. A 243 (1951), 431–462.Google Scholar
3. Barnes, E. S., Non-homogeneous binary quadratic forms, Quarterly J. Math. (2), 1 (1950), 199–210.Google Scholar
4. Chalk, J. H. H., The minimum of a non-homogeneous binary cubic form, Proc. Camb. Phil. Soc. 48 (1952), 392–401.Google Scholar
5. Mordell, L. J., The minima of some inhomogeneous functions of two variables, Duke Math. J. 19 (1952), 519–527.Google Scholar
6. Rogers, K., The minima of some inhomogeneous functions of two variables, J. Lond. Math. Soc. 28 (1953), 394–402.Google Scholar