The dimension of projections of planar diagonal self-affine measures

TL;DR

This paper proves that for certain planar self-affine measures with diagonal linear parts, the Hausdorff dimension of their projections equals the minimum of 1 and their original dimension, under an irrationality condition.

Contribution

It establishes a dimension formula for projections of diagonal self-affine measures under an irrationality assumption, extending understanding of measure projections in fractal geometry.

Findings

Dimension of projected measure equals min(1, original dimension)

Irrationality condition on linear parts is crucial

Results apply to non-principal orthogonal projections

Abstract

We show that if $\mu$ is a self-affine measure on the plane defined by an iterated function system of contractions with diagonal linear parts, then under an irrationality assumption on the entries of the linear parts, $$ \dim_{\rm H} \mu \circ \pi^{-1}= \min\lbrace 1,\dim_{\rm H}\mu\rbrace $$ for any non-principal orthogonal projection $\pi$.