Finer geometry of planar self-affine sets

Abstract

For planar self-affine sets satisfying the strong separation condition, recent work of B\'ar\'any, Hochman, and Rapaport gives mild assumptions under which the Hausdorff dimension equals the affinity dimension. In this paper, we study dominated systems in that regime and ask which finer geometric properties can be characterized. In the range $\dim_{\mathrm{H}}(X) < 1$, we characterize Ahlfors regularity by equivalent conditions involving positivity of $\mathcal{H}^s(X)$, control of projection fibers, and the identity $\dim_{\mathrm{L}}(X)=\dim_{\mathrm{H}}(X)=\dim_{\mathrm{A}}(X)$. In the range $\dim_{\mathrm{H}}(X) \ge 1$, we identify the maximal slice dimension as $\dim_{\mathrm{A}}(X)-1$ in Furstenberg directions and provide examples showing that Marstrand-type all-slice bounds cannot hold in general. We also derive projection consequences for Assouad dimension and exhibit dominated irreducible examples with $\dim_{\mathrm{aff}}(X)<\dim_{\mathrm{A}}(X)$.