Bounded orbits and closed invariant sets for the diagonal group action on $\mathrm{SL}(n,\mathbb{R})/\mathrm{SL}(n,\mathbb{Z})$ and related arithmetic quotients

Let $A$ be the group of positive diagonal matrices acting by right translation on $X_n=\mathrm{SL}(n,\mathbb{R})/\mathrm{SL}(n,\mathbb{Z})$, the space of unimodular lattices in $\mathbb{R}^n$, $n\ge 3$. Lindenstrauss and Weiss (2001) proved that closed $A$-invariant subsets of $X_n$ containing a compact $A$-orbit are homogeneous, under an irrationality hypothesis that is automatic for the standard lattice, and posed the problems this left open. Margulis' original orbit-closure conjecture has since been refuted (Maucourant 2010; Shapira 2011), so the surviving open core is: (1) the Cassels-Swinnerton-Dyer-Margulis bounded-orbit conjecture, asserting that every bounded $A$-orbit in $X_n$ is periodic, equivalently that every compact $A$-invariant subset of $X_n$ contains a compact $A$-orbit, equivalently the minimal-set statements of An-Weiss; a positive answer would imply Littlewood's conjecture on simultaneous Diophantine approximation; (2) a corrected classification of all $A$-orbit closures in $X_n$ and in $G/\Gamma$ for reductive $G$ and arithmetic $\Gamma$, accounting for the known non-homogeneous examples; and (3) the extension of the Lindenstrauss-Weiss homogeneity theorem beyond the standard lattice and beyond $\mathrm{SL}(n,\mathbb{R})$. As of the 2021 survey of Lindenstrauss and the 2023-2025 follow-up literature, all three components remain open.

Background

Let $G=\mathrm{SL}(n,\mathbb{R})$ and let $X_n=G/\mathrm{SL}(n,\mathbb{Z})$ denote the space of unimodular lattices in $\mathbb{R}^n$ (lattices of covolume one), on which $G$ acts by right translation. Let $A\subset G$ be the group of positive diagonal matrices with determinant one, called the diagonal group; it is the identity component of a maximal $\mathbb{R}$-split torus of $G$, meaning a largest connected subgroup of simultaneously diagonalizable matrices. For $n\ge 3$ the group $A$ has rank $n-1\ge 2$, and its action on $X_n$ is the basic example of a higher-rank diagonalizable action. An orbit $A\cdot x$ is called bounded if it is relatively compact in $X_n$, and periodic (or compact) if it is compact, which happens exactly when the stabilizer of $x$ in $A$ is a lattice in $A$. Throughout, a lattice $\Gamma$ in a Lie group means a discrete subgroup of finite covolume, and a subset of $G/\Gamma$ is called homogeneous if it is a finite union of orbits of a closed subgroup $H$ containing $A$.

For unipotent flows, meaning actions of subgroups generated by unipotent elements (matrices all of whose eigenvalues equal 1), Ratner's theorems (1991) completely describe invariant measures and orbit closures: every orbit closure is homogeneous. No comparable general theorem exists for the diagonal group $A$, whose dynamics was highlighted by Furstenberg's 1967 work on the times-2, times-3 map and anticipated by Cassels and Swinnerton-Dyer in their 1955 study of products of three homogeneous linear forms. Margulis (2000) formulated two guiding conjectures: (i) for any closed connected subgroup of a Lie group generated by elements diagonalizable over $\mathbb{R}$ (Ad-split elements), acting on $G/\Gamma$, every orbit closure is either homogeneous or the action factors through that of a one-parameter non-unipotent subgroup; and (ii) the bounded-orbit conjecture, rooted in Cassels-Swinnerton-Dyer, that for $X_n$ with $n\ge 3$ every bounded $A$-orbit is periodic. Margulis proved that conjecture (ii) is equivalent to the hypothesis of Cassels and Swinnerton-Dyer on products of three linear forms, from which Littlewood's conjecture follows. Littlewood's conjecture asserts $\liminf_{n\to\infty} n\,\|n\alpha\|\,\|n\beta\|=0$ for all real $\alpha,\beta$, where $\|\cdot\|$ denotes distance to the nearest integer. The dictionary underlying this implication attaches to a pair $(\alpha,\beta)$ a lattice in $X_3$ whose $A$-orbit stays in a fixed compact set exactly when $(\alpha,\beta)$ admits arbitrarily good simultaneous rational approximations; if all bounded orbits were compact orbits, Littlewood's conjecture would follow.

Lindenstrauss and Weiss (2001) proved the first substantial topological rigidity results in this direction. For $G=\mathrm{SL}(n,\mathbb{R})$, $n\ge 3$, and any lattice $\Gamma$, they showed that a closed $A$-invariant set containing a compact $A$-orbit is homogeneous under a suitable irrationality hypothesis, namely that certain codimension-one subgroups $N_{ij}\subset A$ (the kernels of the characters $a\mapsto a_i/a_j$ of the diagonal group) act ergodically, meaning every invariant measurable set has measure zero or full measure, on the relevant compact $A$-orbits; for $\Gamma=\mathrm{SL}(n,\mathbb{Z})$ this hypothesis is automatic and all such orbit closures are homogeneous. Following unpublished examples of M. Rees for other lattices, they also recorded (in their Section 5) closed $A$-invariant sets that are not homogeneous, built from embedded $\mathrm{SL}(2,\mathbb{R})$ factors, and left open whether a complete classification could be achieved.

Later work reshaped the landscape. Mozes (1995), Tomanov (2002) and Tomanov-Weiss (2003) classified closed $A$-orbits on homogeneous spaces (they correspond to orbits of maximal $\mathbb{R}$-split $\mathbb{Q}$-tori) and extended the framework to $p$-adic and inner-form settings. McMullen (2005) showed that every bounded $A$-orbit closure in $X_n$ meets the well-rounded set, meaning the set of lattices whose shortest nonzero vectors span $\mathbb{R}^n$, and deduced Minkowski's conjecture on products of linear forms in dimension 6. Einsiedler-Katok-Lindenstrauss (2006) classified $A$-invariant measures of positive entropy (entropy measuring the exponential complexity of the measure) and deduced that the set of exceptions to Littlewood's conjecture has Hausdorff dimension zero; this measure-theoretic rigidity leaves the topological bounded-orbit conjecture untouched. Most importantly for the present problem, Margulis' conjecture (i) was refuted: Maucourant (2010) constructed, for $n\ge 6$ and acting by suitable proper subgroups of the diagonal group (both on the standard quotient $X_n$ and on quotients by other lattices), non-homogeneous orbit closures that do not factor through any one-parameter non-unipotent action, and Shapira (2011) produced, for the full diagonal group on $X_3$, irregular orbit closures of a new type. Lindenstrauss-Shapira (2012) distinguished regular points, whose orbit closures are described explicitly, from irregular points, whose orbit closures are finite unions of explicit standard pieces, and noted that "nailing down an explicit conjecture in this direction has proved to be somewhat tricky". An and Weiss (2012) proved, using results of Lindenstrauss-Weiss, that the bounded-orbit conjecture is equivalent to several assertions about minimal sets (nonempty closed invariant sets with no proper closed invariant subset), in particular to the statement that every compact $A$-invariant subset of $X_n$ contains a compact $A$-orbit. Lindenstrauss's 2021 survey states a corrected classification conjecture incorporating a Diophantine growth condition and records that Littlewood's conjecture "remains at present quite open". Recent work of Haynes (2023, withdrawn), Kim (2024) and Kleinbock-Wu (2025) studies bounded orbits and exception sets around these conjectures without settling them.

Problem Statement

Let $n\ge 3$, let $G=\mathrm{SL}(n,\mathbb{R})$, let $A\subset G$ be the group of positive diagonal matrices with determinant one, and let $X_n=G/\mathrm{SL}(n,\mathbb{Z})$ be the space of unimodular lattices in $\mathbb{R}^n$. Determine the closed $A$-invariant subsets and the $A$-orbit closures of $X_n$ and of the related arithmetic quotients, with the following three linked components jointly constituting the objective.

(1) Bounded-orbit (compact-orbit) conjecture of Cassels-Swinnerton-Dyer and Margulis. Prove or refute: every point $x\in X_n$ whose $A$-orbit is bounded (relatively compact) has a periodic, equivalently compact, $A$-orbit. Equivalently (by the equivalences proved by An-Weiss building on Lindenstrauss-Weiss): every compact $A$-invariant subset of $X_n$ contains a compact $A$-orbit. A positive answer would imply Littlewood's conjecture for pairs of real numbers.

(2) Orbit-closure classification. Classify the closures of $A$-orbits in $X_n$, and more generally in $G/\Gamma$ for $G$ a reductive real Lie group and $\Gamma$ an arithmetic lattice (a lattice commensurable with the integer points of a $\mathbb{Q}$-structure on $G$): determine whether a corrected form of Margulis' classification conjecture along the lines of Conjecture 6 of Lindenstrauss's 2021 survey, incorporating the Diophantine growth condition stated there (a subexponential bound on the decay of the shortest nonzero vector along the orbit), holds, where any correct classification accounting for the known non-homogeneous counterexamples of Maucourant (for proper subgroups of the diagonal group) and of Shapira (for the full diagonal group on $X_3$) is admissible; and classify all non-homogeneous orbit closures, determining which arise from the embedded $\mathrm{SL}(2,\mathbb{R})$ (Rees-type) construction of Lindenstrauss-Weiss and which from other mechanisms.

(3) Extension of the homogeneity theorem. Extend the Lindenstrauss-Weiss result that closed $A$-invariant sets containing compact $A$-orbits are homogeneous from the standard lattice $\mathrm{SL}(n,\mathbb{Z})$ to general lattices in semisimple and reductive groups, identifying the sharp compact-orbit and irrationality or ergodicity hypotheses under which the conclusion holds, since Rees-type examples show some such hypothesis is genuinely necessary.

These three components serve one overarching objective, a complete topological classification of the closed invariant sets of the higher-rank diagonal action with its arithmetic consequences, of which component (1) is the arithmetic centerpiece.

References

1. E. Lindenstrauss and B. Weiss, "On sets invariant under the action of the diagonal group", Ergodic Theory and Dynamical Systems 21 (2001), no. 5, 1481-1500. DOI 10.1017/s0143385701001717, https://doi.org/10.1017/s0143385701001717
2. J. W. S. Cassels and H. P. F. Swinnerton-Dyer, "On the product of three homogeneous linear forms and indefinite ternary quadratic forms", Philosophical Transactions of the Royal Society of London, Series A 248 (1955), no. 940, 73-96. DOI 10.1098/rsta.1955.0010, https://doi.org/10.1098/rsta.1955.0010
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16. E. Lindenstrauss, "Recent progress on rigity properties of higher rank diagonalizable actions and applications", arXiv:2101.11114 (2021). https://arxiv.org/abs/2101.11114
17. A. Haynes, "Totally real cubic numbers are well approximable", arXiv:2310.12703 (2023; withdrawn in v2 after a gap was found). https://arxiv.org/abs/2310.12703
18. W. Kim, "On divergent on average trajectories for higher rank actions", arXiv:2403.16559 (2024). https://arxiv.org/abs/2403.16559
19. D. Kleinbock and C. Wu, "Simultaneously bounded and dense orbits for commuting Cartan actions", arXiv:2509.05272 (2025). https://arxiv.org/abs/2509.05272