DS dimension for group actions with infinite orbits

Determine whether dimDS(H_{G,X}) equals the supremum of |A| over all pairs consisting of a subgroup U≤G and a finite set A⊆X that is independent for U and whose U-orbits on A are finite.

Background

The paper proves the upper bound dimDS(H_{G,X})≤H_X(G) without assuming finite orbits and supplies a lower bound from finite-orbit independent sets for subgroups. It gives an infinite-dihedral example where the upper bound is strict.

Equality would characterize PAC learnability for group classes with infinite orbits, including classes arising from discrete groups of isometries.

References

Is $\dimDS(H{G}{X})$ equal to the supremum of $|A|$ over such pairs $(U,A)$? A positive answer would characterize PAC learnability for group classes with infinite orbits, such as discrete groups of isometries.

— Learning a Group Element from Its Action: Galois Correspondences, Optimal ERM, and an Abel-Ruffini Threshold for the Natarajan Dimension  (2610.18154 - Li, 13 Oct 2026) in Section 7, Discussion and open problems, paragraph 4