The Hausdorff dimension of the projections of self-affine carpets

TL;DR

This paper investigates the Hausdorff dimension of projections of self-affine carpets, showing that under certain conditions, the dimension of projections in non-principal directions equals the minimum of the carpet's dimension and 1, generalizing previous results.

Contribution

It extends the understanding of projection dimensions for a broad class of self-affine carpets, including Bedford and McMullen carpets, under irrationality conditions.

Findings

Projections in non-principal directions have Hausdorff dimension min(γ,1).

Results generalize previous work on sums of Cantor sets.

Conditions involve natural irrationality assumptions.

Abstract

We study the orthogonal projections of a large class of self-affine carpets, which contains the carpets of Bedford and McMullen as special cases. Our main result is that if $\Lambda$ is such a carpet, and certain natural irrationality conditions hold, then every orthogonal projection of $\Lambda$ in a non-principal direction has Hausdorff dimension $\min(\gamma,1)$, where $\gamma$ is the Hausdorff dimension of $\Lambda$. This generalizes a recent result of Peres and Shmerkin on sums of Cantor sets.