Projections of self-affine sets onto lines

Balázs Bárány; Antti Käenmäki; István Kolossváry

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arXiv · 2607.14740

Projections of self-affine sets onto lines

Balázs Bárány·Antti Käenmäki·István Kolossváry

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Abstract

We prove an all-directions Marstrand-Mattila projection theorem for self-affine measures and sets in $\mathbb{R}^d$. 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 $X$ itself, without any separation assumption. In the plane, strong irreducibility of the linear parts alone suffices, and this is sharp. As a corollary, if $X$ 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.

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Balázs Bárány, Antti Käenmäki, István Kolossváry. 2026-07-16. Projections of self-affine sets onto lines. https://arxiv.org/abs/2607.14740

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