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

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.

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)
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