Dimension of projected sets and Fourier restriction

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I had a nice discussion with Tuomas after the very nice analysis seminar he gave for the harmonic analysis working group a while ago – he talked about the behaviour of Hausdorff dimension under projection operators and later we discussed the connection with Fourier restriction theory. Turns out there are points of contact but the results one gets are partial, and there are some a priori obstacles.

What follows is an account of the discussion. I will summarize his talk first.

1. Summary of the talk

1.1. Projections in {\mathbb{R}^2}

The problem of interest here is to determine whether there is any drop in the Hausdorff dimension of fractal sets when you project them on a lower dimensional vector space, and if so what can be said about the set of these “bad” projections. This is a very hard problem in general, so one has to start with low dimensions first. In {\mathbb{R}^2} the projections are associated to the points in {\mathbb{S}^1}, namely for {e\in\mathbb{S}^1} one has {\pi_e (x) = (x\cdot e)e}, and so for a given compact set {K} of Hausdorff dimension {0\leq \dim K \leq 1} one asks what can be said about the set of projections for which the dimension is smaller, i.e. {\dim \pi(K) < \dim K}. For {s \leq \dim K}, define the set of directions

\displaystyle E_s (K):= \{e \in \mathbb{S}^1 \,:\, \dim \pi_e (K) < s\}.

We refer to it as to the set of exceptional directions (of parameter {s}). One preliminary result is Marstrand’s theorem:

Theorem 1 (Marstrand) For any compact {K} in {\mathbb{R}^2} s.t. {s<\dim K <1}, one has

\displaystyle |E_s (K)| = 0.

In other words, the dimension is conserved for a.e. direction. The proof of the theorem relies on a characterization of dimension in terms of energy:

Theorem 2 (Frostman’s lemma) For {K} compact in {\mathbb{R}^d}, it is {s<\dim K} if and only if there exists a finite positive Borel measure {\mu} supported in {K} such that

\displaystyle I_s(\mu):= \int_{K}{\int_{K}{\frac{d\mu(x)\,d\mu(y)}{|x-y|^s}}}<\infty.

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