English

Projections of self-affine sets onto lines

Classical Analysis and ODEs 2026-07-16 v1 Dynamical Systems

Abstract

We prove an all-directions Marstrand-Mattila projection theorem for self-affine measures and sets in Rd\mathbb{R}^d. Under exponential separation, together with proximality and strong irreducibility assumptions on the linear parts, the projection of a self-affine measure onto every line has the expected Hausdorff dimension. If the proximality assumption is strengthened to strong pinching, then the same conclusion holds for the self-affine set XX itself, without any separation assumption. In the plane, strong irreducibility of the linear parts alone suffices, and this is sharp. As a corollary, if XX additionally has upper Minkowski dimension at most one, then its Minkowski dimension exists and equals its Hausdorff dimension, giving a partial affirmative answer to the folklore question of whether the Minkowski dimension exists for every self-affine set.

Keywords

Cite

@article{arxiv.2607.14740,
  title  = {Projections of self-affine sets onto lines},
  author = {Balázs Bárány and Antti Käenmäki and István Kolossváry},
  journal= {arXiv preprint arXiv:2607.14740},
  year   = {2026}
}

Comments

69 pages, 1 figure