Minkowski’s conjecture, well-rounded lattices and topological dimension. (English) Zbl 1132.11034

McMullen, Curtis T.

J. Am. Math. Soc. 18, No. 3, 711-734 (2005).

One of the great problems of the geometry of numbers is the following conjecture on the product of non-homogeneous linear forms: For \(x\in \mathbb R^{n}\) let \(| x| ^{2}=x_1^{2}+\cdots +x_n^{2}\) and \(N(x)=| x_1\cdots x_n| \). Then for any lattice \(L\) in \(\mathbb R^{n}\) with determinant 1 holds
\[ \sup_{x\in\mathbb R^{n}} \inf_{y\in L} N(x-y)\leq 2^{-n}. \]
Equality holds if and only if \(L=D\mathbb Z^{n}\) where \(D\) is a diagonal matrix with positive entries and determinant 1. The conjecture has been proved for \(n=2,3,4,5\) by [M. Minkowski, Diophantische Approximation. Neudruck. Würzburg: Physica-Verlag (1961; Zbl 0103.03403)], R. Remak [Math. Z. 17, 1–34, 18, 173–200 (1923; JFM 49.0101.03)], F. J. Dyson [Ann. Math. (2) 49, 82–109 (1948; Zbl 0031.15402)] and B. F. Skubenko [Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 33, 6–36 (1973; Zbl 0352.10010)]. For general \(n\) there are estimates due to N. G. Chebotarev [Vierteljahresschr. Naturforsch. Ges. Zürich 85, Beibl. 32, 27–30 (1940; Zbl 0023.20701)] and others including E. Bombieri [Acta Math. 8, 273–281 (1963; Zbl 0123.04505)]. For large sets of lattices the conjecture has been proved by A. M. Macbeath [Proc. Glasg. Math. Assoc. 5, 86–89 (1961; Zbl 0098.26303)] and the reviewer [Acta Arith. 13, 9–27 (1967; Zbl 0153.07102)]. One line of attack is to show the following:
(i) For each lattice \(L\) in \(\mathbb R^{n}\) there is a diagonal matrix \(D\) as above such that \(DL\) has the following property: the set of vectors \(y\in L\backslash\{ o\}\) of minimum length \(| y| \) span \(\mathbb R^{n}\), i.e. \(DL\) is well rounded.
(ii) If \(L\) is well rounded and has determinant 1, then its covering radius satisfies \[ \sup\limits_{x\in\mathbb R^{n}} \inf_{y\in L} | x-y| \leq {\sqrt{n}\over 2} \]
where equality holds only in a particular case.
If (i) and (ii) hold, the inequality of the arithmetic and geometric mean yields the conjecture. For more information see the reviewer and C. G. Lekkerkerker [Geometry of numbers. 2nd ed., North-Holland Mathematical Library, Vol. 37. Amsterdam etc.: North-Holland (1987; Zbl 0611.10017)] and the reviewer [Convex and discrete geometry. Grundlehren der Mathematischen Wissenschaften 336. Berlin: Springer (2007; Zbl 1139.52001)].
Proposition (ii) has been proved by A. C. Woods [J. Number Theory 4, 157–180 (1972; Zbl 0232.10020)] for \(n=6\). The author shows (i) for all lattices of determinant 1 with \(N(L)>0\) in all dimensions. Using a result of B. J. Birch and H. P. F. Swinnerton-Dyer [Mathematika Lond. 3, 25–39 (1956; Zbl 0074.03702)] this finally yields the conjecture for \(n=6\). This result is an important contribution to the geometry of numbers.

Reviewer: Peter M. Gruber (Wien)

MSC:

11H46	Products of linear forms
11H31	Lattice packing and covering (number-theoretic aspects)
11J20	Inhomogeneous linear forms
52C17	Packing and covering in \(n\) dimensions (aspects of discrete geometry)

Keywords:

conjecture on inhomogeneous linear forms; lattice covering with balls

Citations:

Zbl 0103.03403; Zbl 0031.15402; Zbl 0352.10010; Zbl 0023.20701; Zbl 0123.04505; Zbl 0098.26303; Zbl 0153.07102; Zbl 0611.10017; Zbl 0232.10020; Zbl 0074.03702; Zbl 1139.52001; JFM 49.0101.03

References:

[1]	Avner Ash, Small-dimensional classifying spaces for arithmetic subgroups of general linear groups, Duke Math. J. 51 (1984), no. 2, 459 – 468. · Zbl 0542.22011
[2]	Avner Ash and Mark McConnell, Cohomology at infinity and the well-rounded retract for general linear groups, Duke Math. J. 90 (1997), no. 3, 549 – 576. · Zbl 0903.11016
[3]	R. P. Bambah and A. C. Woods, Minkowski’s conjecture for \?=5; a theorem of Skubenko, J. Number Theory 12 (1980), no. 1, 27 – 48. · Zbl 0426.10030
[4]	Eva Bayer-Fluckiger, Lattices and number fields, Algebraic geometry: Hirzebruch 70 (Warsaw, 1998) Contemp. Math., vol. 241, Amer. Math. Soc., Providence, RI, 1999, pp. 69 – 84. · Zbl 0951.11016
[5]	Eva Bayer-Fluckiger, Ideal lattices, A panorama of number theory or the view from Baker’s garden (Zürich, 1999) Cambridge Univ. Press, Cambridge, 2002, pp. 168 – 184. · Zbl 1043.11057
[6]	E. Bayer-Fluckiger and G. Nebe. On the Euclidean minimum of some real number fields. Preprint, 2004. · Zbl 1161.11032
[7]	B. J. Birch and H. P. F. Swinnerton-Dyer, On the inhomogeneous minimum of the product of \? linear forms, Mathematika 3 (1956), 25 – 39. · Zbl 0074.03702
[8]	A. I. Borevich and I. R. Shafarevich, Number theory, Translated from the Russian by Newcomb Greenleaf. Pure and Applied Mathematics, Vol. 20, Academic Press, New York-London, 1966. · Zbl 0145.04902
[9]	Raoul Bott and Loring W. Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics, vol. 82, Springer-Verlag, New York-Berlin, 1982. · Zbl 0496.55001
[10]	J. W. S. Cassels and H. P. F. Swinnerton-Dyer, On the product of three homogeneous linear forms and the indefinite ternary quadratic forms, Philos. Trans. Roy. Soc. London. Ser. A. 248 (1955), 73 – 96. · Zbl 0065.27905
[11]	J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, 3rd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 290, Springer-Verlag, New York, 1999. With additional contributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov. · Zbl 0915.52003
[12]	H. Davenport. A simple proof of Remak’s theorem on the product of \(n\) linear forms. J. London Math. Soc. 14(1939), 47-51. · Zbl 0020.20502
[13]	F. J. Dyson, On the product of four non-homogeneous linear forms, Ann. of Math. (2) 49 (1948), 82 – 109. · Zbl 0031.15402
[14]	Samuel Eilenberg and Norman Steenrod, Foundations of algebraic topology, Princeton University Press, Princeton, New Jersey, 1952. · Zbl 0047.41402
[15]	Michael Gromov, Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. 56 (1982), 5 – 99 (1983). · Zbl 0516.53046
[16]	P. M. Gruber and C. G. Lekkerkerker, Geometry of numbers, 2nd ed., North-Holland Mathematical Library, vol. 37, North-Holland Publishing Co., Amsterdam, 1987. · Zbl 0611.10017
[17]	Witold Hurewicz and Henry Wallman, Dimension Theory, Princeton Mathematical Series, v. 4, Princeton University Press, Princeton, N. J., 1941. · Zbl 0060.39808
[18]	J. F. Koksma, Diophantische Approximationen, Springer-Verlag, Berlin-New York, 1974 (German). Reprint. · Zbl 0276.10015
[19]	Elon Lindenstrauss and Barak Weiss, On sets invariant under the action of the diagonal group, Ergodic Theory Dynam. Systems 21 (2001), no. 5, 1481 – 1500. · Zbl 1073.37006
[20]	Gregory Margulis, Problems and conjectures in rigidity theory, Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 161 – 174. · Zbl 0952.22005
[21]	Hermann Minkowski, Diophantische Approximationen. Eine Einführung in die Zahlentheorie, Chelsea Publishing Co., New York, 1957 (German). · JFM 53.0165.01
[22]	Hee Oh, Finiteness of compact maximal flats of bounded volume, Ergodic Theory Dynam. Systems 24 (2004), no. 1, 217 – 225. · Zbl 1048.22002
[23]	Pierre Pansu, Introduction to \?² Betti numbers, Riemannian geometry (Waterloo, ON, 1993) Fields Inst. Monogr., vol. 4, Amer. Math. Soc., Providence, RI, 1996, pp. 53 – 86. · Zbl 0848.53025
[24]	Дискретные подгруппы групп Ли., Издат. ”Мир”, Мосцощ, 1977 (Руссиан). Транслатед фром тхе Енглиш бы О. В. Šварцман; Едитед бы Ѐ. Б. Винберг; Щитх а супплемент ”Аритхметициты оф ирредуцибле латтицес ин семисимпле гроупс оф ранк греатер тхан 1” бы Г. А. Маргулис.
[25]	R. Remak. Verallgemeinerung eines Minkowskischen Satzes. Math. Zeitschr. 17-18(1928), 1-34; 173-200.
[26]	B. F. Skubenko, A new variant of the proof of the inhomogeneous Minkowski conjecture for \?=5, Trudy Mat. Inst. Steklov. 142 (1976), 240 – 253, 271 (Russian). Number theory, mathematical analysis and their applications. · Zbl 0415.10021
[27]	B. F. Skubenko. A proof of Minkowski’s conjecture on the product of \(n\) linear inhomogeneous forms in \(n\) variables for \(n\leq 5\). J. Soviet Math. 6(1976), 627-650. · Zbl 0374.10018
[28]	Christophe Soulé, The cohomology of \?\?\(_{3}\)(\?), Topology 17 (1978), no. 1, 1 – 22. · Zbl 0382.57026
[29]	George Tomanov and Barak Weiss, Closed orbits for actions of maximal tori on homogeneous spaces, Duke Math. J. 119 (2003), no. 2, 367 – 392. · Zbl 1040.22005
[30]	P. Vámos, The missing axiom of matroid theory is lost forever, J. London Math. Soc. (2) 18 (1978), no. 3, 403 – 408. · Zbl 0395.05024
[31]	A. C. Woods, Covering six space with spheres, J. Number Theory 4 (1972), 157 – 180. · Zbl 0232.10020