The Nikodym dimension conjecture for constant-curvature manifolds

Let $(M^d,g)$ be a Riemannian manifold of dimension $d$ and constant sectional curvature, and let a Nikodym set be a Borel set whose associated Nikodym enlargement has positive measure for all parameters sufficiently close to $1$. Nikodym dimension conjecture. The Hausdorff dimension of a Nikodym set is $d$. The conjecture is known in dimension three with constant sectional curvature by the results described in the paper, while the general-dimensional statement remains open.