Estimating the critical determinants of a class of three-dimensional star bodies

Open access

Abstract

In the problem of (simultaneous) Diophantine approximation in ℝ3 (in the spirit of Hurwitz’s theorem), lower bounds for the critical determinant of the special three-dimensional body K2 : (y2 + z2)(x2 + y2 + z2) ≤ 1 play an important role; see [1], [6]. This article deals with estimates from below for the critical determinant ∆ (Kc) of more general star bodies Kc : (y2 + z2)c/2(x2 + y2 + z2) ≤ 1 ; where c is any positive constant. These are obtained by inscribing into Kc either a double cone, or an ellipsoid, or a double paraboloid, depending on the size of c.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1] J.V. Armitage: On a method of Mordell in the geometry of numbers. Mathematika 2 (2) (1955) 132-140.

  • [2] H. Davenport K. Mahler: Simultaneous Diophantine approximation. Duke Math. J. 13 (1946) 105-111.

  • [3] H. Davenport: On a theorem of Furtwängler. J. London Math.Soc. 30 (1955) 185-195.

  • [4] P.M. Gruber C.G. Lekkerkerker: Geometry of numbers. North Holland Amsterdam (1987).

  • [5] H. Minkowski: Dichteste gitterförmige Lagerung kongruenter Körper. Nachr. Kön. Ges. Wiss. Göttingen (1904) 311-355.

  • [6] W.G. Nowak: The critical determinant of the double paraboloid and Diophantine approximation in R3 and R4. Math. Pannonica 10 (1999) 111-122.

  • [7] W.G. Nowak: Diophantine approximation in Rs: On a method of Mordell and Armitage. In: Algebraic number theory and Diophantine analysis. Proceedings of the conference held in Graz Austria August 30 to September 5 1998 W. de Gruyter Berlin. (2000) 339-349.

  • [8] W.G. Nowak: Lower bounds for simultaneous Diophantine approximation constants. Comm. Math. 22 (1) (2014) 71-76.

  • [9] W.G. Nowak: Simultaneous Diophantine approximation: Searching for analogues of Hurwitz's theorem. In: T.M. Rassias and P.M. Pardalos (eds.) Essays in mathematics and its applications. Springer Switzerland (2016) 181-197.

  • [10] W.G. Nowak: On the critical determinants of certain star bodies. Comm. Math. 25 (1) (2017) 5-11.

  • [11] K. Ollerenshaw: The critical lattices of a sphere. J. London Math. Soc. 23 (1949) 297-299.

  • [12] J.V. Whitworth: The critical lattices of the double cone. Proc. London Math. Soc. 2 (1) (1951) 422-443.

Search
Journal information
Impact Factor


CiteScore 2018: 0.4

SCImago Journal Rank (SJR) 2018: 0.193
Source Normalized Impact per Paper (SNIP) 2018: 0.696

Mathematical Citation Quotient (MCQ) 2018: 0.17

Target audience:

researchers in all areas of pure and applied mathematics

Metrics
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 323 175 9
PDF Downloads 133 79 0