On the dimension of orthogonal projections of self-similar measures

Amir Algom; Pablo Shmerkin

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arXiv · 2407.16262

On the dimension of orthogonal projections of self-similar measures

Amir Algom·Pablo Shmerkin

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Abstract

Let $\nu$ be a self similar measure on $\mathbb{R}^d$, $d\geq 2$, and let $\pi$ be an orthogonal projection onto a $k$-dimensional subspace. We formulate a criterion on the action of the group generated by the orthogonal parts of the IFS on $\pi$, and show that it ensures the dimension of $\pi \nu$ is preserved; this significantly refines previous results by Hochman-Shmerkin (2012) and Falconer-Jin (2014), and is sharp for projections to lines and hyperplanes. A key ingredient in the proof is an application of a restricted projection theorem of Gan-Guo-Wang (2024).

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Amir Algom, Pablo Shmerkin. 2024-07-23. On the dimension of orthogonal projections of self-similar measures. https://arxiv.org/abs/2407.16262

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