This question came up when my supervisors, my colleague, and I were considering arithmetic progressions in sets of fractional dimension. In particular, we were interested in "extracting" Salem sets from other sets, and we came across the following question:

We define the *Fourier dimension* of $E \subseteq \mathbb{R}$, $\mathrm{dim}_F(E)$, as the supremum of $\beta \in [0,1]$ such that for some probability measure $\mu$ supported on $E$,

$ |\widehat{\mu}(\xi)| \leq C|\xi|^{-\beta/2}. $

**Suppose $E_1,E_2$ are two subsets of $\mathbb{R}$. What is the relationship, if any, between $\mathrm{dim}_F(E_1 + E_2)$ and $\mathrm{dim}_F(E_1)$, $\mathrm{dim}_F(E_2)$?**

A quick calculation shows

$ \mathrm{dim}_F(E_1 + E_2) \geq \min(\mathrm{dim}_F(E_1) + \mathrm{dim}_F(E_2),1), $

but can we do any better than this? Or can strict inequality hold?

For some motivation, we may look to Hausdorff dimension instead (it is known that $\mathrm{dim}_F(E) \leq \mathrm{dim}_H(E)$; a set $E$ such that $\mathrm{dim}_F(E) = \mathrm{dim}_H(E)$ is called a *Salem set*). I believe Falconer gave an example of sets $E_1$ and $E_2$ such that $\mathrm{dim}_H(E_1) = 0 = \mathrm{dim}_H(E_2)$, yet $\mathrm{dim}_H(E_1+E_2) = 1$.