Abstract
For a latticeL in ℝn with determinantd(L), let η (L) denote the supremum of the values 2−2 V(P)/d(L), taken over theL-admissible parallelepidesP, symmetric with respect to the origin and with faces parallel to the coordinate-axes. In 1936, Mordell asked for the constants ℵ n = min ℵ(L) over alln-dimensional lattices. In this paper we investigate isolated minima of η (L) in all over alln-dimensional lattices. In this paper we (Satz 1) and some examples are given. In particular, forn<=4, the set of lattices with isolated η turns out to be dense in the space of lattices. Conversely, the set of (algebraically generated) lattices with non-isolated η is dense, at least in the case of a plane (Satz 2).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Literatur
Bambah, R. P., Woods, A. C.: On a theorem of Dyson. J. Number Th.6, 422–433 (1974).
Barnes, E. S.: Isolated minima of the product ofn linear forms. Proc. Camb. Phil. Soc.49, 59–62 (1953).
Bullig-Bergmann, G.: Ein periodisches Verfahren zur Berechnung eines Systems von Grundeinheiten in total reellen kubischen Körpern. Abh. Math. Sem. Hamburg12, 369–414 (1938).
Cassels, J. W. S., Swinnerton-Dyer, H.P.F.: On the product of three homogeneous linear forms and indefinite ternary quadratic forms. Phil. Trans. Royal Soc. London A248, 73–96 (1955).
Gruber, P.: Bemerkungen zum Umkehrproblem für den Minkowskischen Linearformensatz. Ann. Univ. Sci. Budapest13, 5–10 (1970).
Gruber, P.: Geometry of Numbers. In: Beiträge zur Geometrie. Proc. Geom. Symp. Siegen 1978. Basel-Stuttgart: Birkhäuser. 1978.
Gruber, P., Ramharter, G.: Beiträge zum Umkehrproblem für den Minkowskischen Linearformensatz. Acta Math. Acad. Sci. Hung. (Im Druck.)
Hlawka, E.: Über Gitterpunkte in Parallelepipeden. J. reine angew. Math.87, 246–252 (1951).
Ko, C.: Note on lattice points in a parallelepiped. J. Lond. Math. Soc.12, 40–47 (1937).
Lekkerkerker, G. G.: Geometry of Numbers. Groningen: Wolters-Noordhoff. Amsterdam North-Holland. 1969.
Minkowski, H.: Gesammelte Abhandlungen, Bd. 1. Leipzig-Berlin: Teubner. 1911.
Mordell, L.J.: Note on an arithmetical problem on linear forms. J. Lond. Math. Soc.12, 34–36 (1937).
Noodrzij, P.: Über das Produkt von vier reellen homogenen linearen Formem. Mh. Math.71, 436–445 (1967).
Oppenheim, A.: The continued fractions associated with chains of quadratic forms. Proc. London Math. Soc. (2)44, 323–335 (1937).
Ramharter, G.: Über das Umkehrproblem für den Minkowskischen Linearformensatz. Acta Arithm.36, 27–41 (1980).
Swinnerton-Dyer, H. P. F.: On the product of three homogeneous linear forms. Acta Arithm.18, 371–385 (1971).
Suranyi, J.: Über einen Satz von G. Szekeres in der Geometrie der Zahlen. Ann. Univ. Sci. Budapest.3–4, 319–326 (1960/61).
Szekeres, G.: On a problem of the lattice plane. J. Lond. Math. Soc.12, 88–93 (1937).
Szekeres, G.: Note on lattice points in a parallelepiped. J. Lond. Math. Soc.12, 36–39 (1937).
Szüsz, P.: Beweis eines zahlengeometrischen Satzes von Szekeres. Acta Math. Acad. Sci. Hung.7, 75–79 (1956).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Ramharter, G. Über ein Problem von Mordell in der Geometrie der Zahlen. Monatshefte für Mathematik 92, 143–160 (1981). https://doi.org/10.1007/BF01303745
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01303745