Margulis's conjecture on bounded diagonal orbits in the space of lattices, and the Woods covering conjecture for well-rounded lattices in dimensions 11-23

McMullen's paper 'Minkowski's conjecture, well-rounded lattices and topological dimension' (J. Amer. Math. Soc. 18 (2005) 711-734) introduced a dynamical approach to Minkowski's conjecture on the product of $n$ linear forms - the assertion that every unimodular lattice $L \subset \mathbb{R}^n$ satisfies $\sup_x \inf_{y \in L} |(x_1-y_1)\cdots(x_n-y_n)| \le 2^{-n}$ - and recorded four open threads, two of which remain open. This task is to settle those two: Margulis's conjecture (Conjecture 3.2 of the paper), that for $n \ge 3$ a diagonal-group orbit with compact closure is already compact - equivalently, that every lattice with positive homogeneous minimum $N(L) > 0$ arises from a totally real number field - and the Woods covering conjecture $(C_k)$ for $11 \le k \le 23$, each dimension of which would extend Minkowski's conjecture beyond the known range $n \le 10$. The other two threads are described as context in the Background.

Background

This problem records the surviving open core of the questions left by McMullen's paper on Minkowski's conjecture, well-rounded lattices and topological dimension [1], which approaches a century-old problem of the geometry of numbers through the dynamics of the diagonal group on the space of lattices.

Minkowski's conjecture. A lattice $L \subset \mathbb{R}^n$ is a discrete subgroup of rank $n$; it is unimodular when its fundamental domain has Euclidean volume 1. For $x = (x_1,\dots,x_n)$ write $N(x) = |x_1 x_2 \cdots x_n|$, the product of the coordinates. Minkowski's conjecture on the product of linear forms asserts that every unimodular lattice is coarse enough to approximate every point with error at most $2^{-n}$ in this product norm:

$$\sup_{x \in \mathbb{R}^n} \; \inf_{y \in L} N(x-y) \;\le\; 2^{-n}.$$

It was settled for $n \le 5$ through work of Remak, Dyson, Skubenko, Birch-Swinnerton-Dyer and others [2],[4]: in particular, Birch and Swinnerton-Dyer showed that a counterexample of minimal dimension must have $N(L) = \inf\{N(y) : y \in L,\, y \ne 0\} > 0$ (positive homogeneous minimum). The source paper [1] pushed this to $n = 6$, and subsequent work verified $n = 7, 8$ [10],[11], $n = 9$ [12], and $n = 10$ [18]. The conjecture is open for every $n \ge 11$.

The dynamical framework. The space of unimodular lattices in $\mathbb{R}^n$ is identified with the homogeneous space $X_n = SL_n(\mathbb{R})/SL_n(\mathbb{Z})$. Let $A \subset SL_n(\mathbb{R})$ be the group of positive diagonal matrices of determinant 1. By a compactness criterion of Mahler type, the orbit $A \cdot L$ is bounded (relatively compact) in $X_n$ exactly when $N(L) > 0$. Theorem 3.1 of [1] characterizes compact orbits: $A \cdot L$ is compact if and only if, up to homothety and the action of $A$, the lattice $L$ is the image of a full module in a totally real number field of degree $n$ (a module embedded via the $n$ real embeddings of the field), if and only if $N(L) > 0$ and the set $\{N(y) : y \in L\}$ is discrete in $\mathbb{R}$. Thus 'bounded orbit' is a Diophantine condition, while 'closed orbit' is an arithmetic one.

Margulis's conjecture. Margulis (in his 2000 problem list, cited as Conjecture 3.2 in [1]) conjectured that for $n \ge 3$, every $A$-orbit with compact closure is already compact (equivalently closed). By Theorem 3.1 this is equivalent to: every lattice with $N(L) > 0$ arises from a totally real number field. The case $n = 3$ implies Littlewood's conjecture - $\liminf_{m \to \infty} m \,\|m\alpha\|\, \|m\beta\| = 0$ for all real $\alpha, \beta$, where $\|t\|$ denotes the distance from $t$ to the nearest integer - which is the central open problem of simultaneous Diophantine approximation and has been open since the 1930s. The strongest evidence is the measure-theoretic counterpart of Einsiedler-Katok-Lindenstrauss [7]: $A$-invariant ergodic measures on $X_3$ with positive entropy are algebraic, whence the set of exceptions to Littlewood's conjecture has Hausdorff dimension zero. However, a bounded orbit can support invariant measures of zero entropy, precisely where measure rigidity gives no information, so the topological statement is beyond current techniques. Related classifications stop short of it: Lindenstrauss-Weiss [5] show that any orbit closure containing a compact orbit is a homogeneous submanifold; Tomanov-Weiss [6] classify closed orbits for actions of maximal tori; Maucourant [8] shows that the analogous topological rigidity fails for proper subgroups of the diagonal acting on quotients by non-arithmetic lattices (which does not touch the full diagonal on $SL_n(\mathbb{Z})$); and Lindenstrauss-Shapira [9] construct further orbit-closure phenomena in $X_d$ for $d > 2$, all with unbounded orbits.

The covering route. A lattice is well-rounded if its minimal vectors (the nonzero vectors of smallest Euclidean norm) span $\mathbb{R}^n$; its Euclidean covering radius is $\mathrm{cov}(L) = \sup_{x \in \mathbb{R}^n} \inf_{y \in L} |x - y|$, the least $r$ such that balls of radius $r$ centered at lattice points cover $\mathbb{R}^n$. By the arithmetic-geometric mean inequality, $N(x)^{1/n} \le |x|/\sqrt{n}$, so Minkowski's conjecture in dimension $n$ would follow if every lattice could be moved by the diagonal group into the well-rounded locus, where the covering radius is at most $\sqrt{n}/2$. McMullen [1] proved the dynamical half: any lattice whose diagonal orbit has compact closure has a well-rounded lattice in its orbit closure; combined with the Birch-Swinnerton-Dyer reduction, this yields Corollary 1.4 of [1]: if the covering conjectures $(C_k)$ hold for all $k \le n$ - where $(C_k)$ asserts that every well-rounded unimodular lattice $L \subset \mathbb{R}^k$ satisfies $\mathrm{cov}(L) \le \sqrt{k}/2$, with equality only for rotates $g \cdot \mathbb{Z}^k$, $g \in SO_k(\mathbb{R})$ - then Minkowski's conjecture holds in all dimensions $\le n$. Woods [3] verified $(C_k)$ for $k \le 6$; later work verified $k = 7, 8$ [10],[11], $k = 9$ [12], and $k = 10$ [18] (hence Minkowski for $n \le 10$). Regev-Shapira-Weiss [15] then disproved $(C_k)$ in general: the supremum $N_d$ of covering radii of well-rounded unimodular lattices in $\mathbb{R}^d$ satisfies $N_d > c\,d/\sqrt{\log d}$ for an absolute constant $c > 0$, and in particular $N_d > \sqrt{d}/2$ for every $d \ge 30$; Chen-Xu [16] extended the refutation to every $d \ge 24$. The truth of $(C_k)$ in the remaining range $11 \le k \le 23$ is unknown, and the same paper's lattices do not refute Minkowski's conjecture itself. Shapira-Weiss [13] proposed an alternative route through stable lattices (a unimodular lattice is stable if every nonzero primitive sublattice has covolume at least 1): every orbit closure meets the stable locus, and the analogous covering conjecture for stable lattices would also imply Minkowski's conjecture. Solan [17] proved the definitive structural statement: every diagonal orbit itself meets both the stable locus and the well-rounded locus, for any choice of norm.

The two further recorded directions. The source paper records two additional open questions. First, the well-rounded lattices form a spine (deformation retract) for the quotient $SO_n(\mathbb{R})\backslash SL_n(\mathbb{R})/SL_n(\mathbb{Z})$, and [1] asks for 'a more complete description of the way geodesic flats can meet the well-rounded spine in higher dimensions'; the existence aspect is now fully resolved [13],[14],[17], while the finer geometric description remains an open-ended direction without a determinate completion boundary. Second, [1] asks which families of coordinate subsets arise as supports of measured subspaces $M \subset \mathbb{R}^n$ (the nonzero Plücker coordinates of the volume element of $M$); these are exactly the bases of real-representable matroids, and the finite-axiomatization reading of this question, posed by Vámos [19], was answered negatively by Mayhew-Newman-Whittle [20]: no sentence of monadic second-order logic characterizes representability over any infinite field. The structural characterization of real-representable matroids remains a major open problem in matroid theory, but it is not part of the dynamical framework of this problem.

Problem Statement

This problem is the surviving precise open core of the four open threads recorded in [1]. It has two linked components, both arising from the dynamical framework of [1] for Minkowski's conjecture. The common objective is to complete the arithmetic classification of bounded diagonal orbits and to determine how far the covering conjectures $(C_k)$ extend Minkowski's conjecture; the two components are independent.

Component (i) - Margulis's conjecture (Conjecture 3.2 of [1]). For every $n \ge 3$: prove or refute that every orbit $A \cdot L$ of the diagonal group $A$ in the space of lattices $X_n = SL_n(\mathbb{R})/SL_n(\mathbb{Z})$ whose closure is compact is itself compact (equivalently, closed). Equivalently: prove or refute that every unimodular lattice $L \subset \mathbb{R}^n$ with $N(L) = \inf\{ |y_1 y_2 \cdots y_n| : y \in L,\, y \ne 0\} > 0$ arises, up to homothety and the action of $A$, from a full module in a totally real number field of degree $n$. The case $n = 3$ implies Littlewood's conjecture, and a single counterexample in any $n \ge 3$ refutes the conjecture as stated.

Component (ii) - Woods's covering conjecture in the remaining open range. For each $k$ with $11 \le k \le 23$: prove or refute $(C_k)$, the statement that every well-rounded unimodular lattice $L \subset \mathbb{R}^k$ (minimal vectors spanning $\mathbb{R}^k$) has Euclidean covering radius $\mathrm{cov}(L) = \sup_{x \in \mathbb{R}^k} \inf_{y \in L} |x - y|$ at most $\sqrt{k}/2$, with equality only for lattices $g \cdot \mathbb{Z}^k$ with $g \in SO_k(\mathbb{R})$. By Corollary 1.4 of [1], a proof of $(C_k)$ for all $k \le n$ implies Minkowski's conjecture in all dimensions $\le n$; since $(C_k)$ is already known for $k \le 10$ [3],[10],[11],[12],[18], a proof of $(C_k)$ for every $k$ from $11$ up to a given dimension $n$ with $11 \le n \le 23$ extends Minkowski's conjecture (currently known only for $n \le 10$) to that dimension - in particular, a proof of $(C_{11})$ alone extends it to $n = 11$. A refutation at some $k$ in this range would show that the classical Remak-Davenport-Woods route fails below the currently known threshold $k = 24$ [15],[16].

The two components are recorded separately in [1] and are independent: component (i) is refuted by a counterexample in some $n \ge 3$ and proved for every $n \ge 3$, and component (ii) is decided (proved or refuted) for every $k$ with $11 \le k \le 23$. Proving Minkowski's conjecture in a dimension $n \ge 11$ does not by itself establish $(C_k)$.

References

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