Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-06T02:11:23.574Z Has data issue: false hasContentIssue false

A Restricted Inhomogeneous Minimum for Forms

Published online by Cambridge University Press:  09 April 2009

P. E. Blanksby
Affiliation:
Mathematics Department University of Adelaide
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let us suppose that ƒ(x, y) is an indefinite binary quadratic form that does not represent zero. If P is the real point (x0, y0) then we may define where the infimum is taken over all integral x, y. The inhomogeneous minimum of the form ƒ is defined where the supremum taken over all real points P, need only extend over some complete set of points, incongruent mod 1.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1969

References

[1]Barnes, E. S., ‘The inhomogeneous minima of binary quadratic forms IV’, Acta Math. 92 (1954), 235264.CrossRefGoogle Scholar
[2]Barnes, E. S., ‘On linear inhomogeneous diophantine approximation’, J. London Math. Soc. 31 (1956), 7379.CrossRefGoogle Scholar
[3]Barnes, E. S., and Swinnerton-Dyer, H. P. F., ‘The inhomogeneous minima of binary quadratic forms III’, Acta Math. 92 (1954), 199234.CrossRefGoogle Scholar
[4]Blanksby, P. E., ‘On the product of two linear forms, one homogeneous and one inhomogeneous’, J. Aust. Math. Soc. 8 (1968), 457511.CrossRefGoogle Scholar
[5]Blanksby, P. E., Ph. D. Thesis (submitted to University of Adelaide, Australia, 1967).Google Scholar
[6]Cassels, J. W. S., ‘Über ’, Math. Ann. 127 (1954) 288304.CrossRefGoogle Scholar
[7]Descombes, R., ‘Sur la répartition des sommets d'une ligne polygonale régulière non fermée’, Ann. Sci. Ec. Norm. Sup. 75 (1956), 284355.Google Scholar
[8]Descombes, R., ‘Problèmes d'approximation diophantienne’, L'Enseignement Math. 6 (1960), 1826.Google Scholar
[9]Dickson, L. E., Introduction to the Theory of Numbers (Dover, 1957).Google Scholar
[10]Pitman, E. J. Ph. D. Thesis (University of Sydney, Australia, 1957).Google Scholar
[11]Pitman, E. J., ‘The inhomogeneous minima of a sequence of symmetric Markov forms’, Acta Arith., 5 (1958), 81116.CrossRefGoogle Scholar
[12]Pitman, E. J., ‘Davenport's Constant for indefinite binary quadratic forms’, Acta Arith., 6 (1960), 3746.CrossRefGoogle Scholar