J. Marklof

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For $d\geq1$, let $\mathbf{Cl}(\mathbb R^d)$ be the space of all closed subsets of $\mathbb R^d$, including $\varnothing$ and $\mathbb R^d$. Its Chabauty--Fell topology is generated by the conditions that a closed set meet a specified open set and avoid a specified compact set. Let $\mathrm{SL}_d(\mathbb R)$ be the real determinant-one linear maps and $\mathrm{ASL}_d(\mathbb R)=\mathrm{SL}_d(\mathbb R)\ltimes\mathbb R^d$ their affine extensions. They act by $F\mapsto gF$. A Borel probability measure $\mu$ is invariant if $\mu(g^{-1}A)=\mu(A)$ for every Borel $A$ and every group element $g$.

Statement recorded in the supplied source.

Classify all $\mathrm{SL}_d(\mathbb R)$-invariant Borel probability measures on $\mathbf{Cl}(\mathbb R^d)$, and likewise all $\mathrm{ASL}_d(\mathbb R)$-invariant Borel probability measures. [S2]

Short English statement

Describe every invariant random closed set for the linear and affine volume-preserving groups.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 6600019 (AMR-065-0019). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Faustin Adiceam (compiler), Open Problems and Conjectures related to the Theory of Mathematical Quasicrystals, arXiv:1604.06280v2 (2016), Section 2.8.1, Problem 2.8.1, pp. 10--11. https://arxiv.org/pdf/1604.06280.