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 $\mathrm{dim}\_H(E\_1+E\_2) = 1$.



Fourier dimension of the sum of sets
It is possible that $\dim\_F(E\_1)=\dim\_F(E\_2)=0$ yet $\dim\_F(E\_1+E\_2)=1$, so there is no inequality in the opposite direction. 


In fact, Falconer's example of sets $E\_1, E\_2$ such that $\dim\_H(E\_1)=\dim\_H(E\_2)=0$ but $\dim\_H(E\_1+E\_2)=1$ already works. In Falconer's example, not only $E\_1+E\_2$ has dimension $1$, but in fact $E\_1+E\_2$ is an interval. Hence $\dim\_F(E\_1)=\dim\_F(E\_2)=0$ (since $\dim\_F$ is bounded above by $\dim\_H$) but $\dim\_F(E\_1+E\_2)=1$.


There are also examples which are not forced by the Hausdorff dimensions of the sets. Indeed, let $C$ be the ternary Cantor set. It is a classical result of Kahane and Salem that if $\mu$ is any measure supported on $C$, then $\widehat{\mu}(\xi)\nrightarrow 0$ as $|\xi|\to\infty$; in particular, $\dim\_F(C)=0$. Clearly, the same is true for any dilate $t C$ with $t\neq 0$.


On the other hand, it is well known that $C+C$ equals the interval $[0,2]$. Indeed, since $C$ and also $tC$ have Newhouse thickness equal to $1$, Newhouse's gap lemma (the endpoint version), guarantees that $C+tC$ has nonempty interior for all $t\neq 0$, and therefore also Fourier dimension $1$ (note that I'm not claiming that the indicator function of $C+tC$ has Fourier decay, just that there is a measure supported on (an interval in) $C+tC$ that does).


Hence, for all $t$ we have $\dim\_F(C)=\dim\_F(tC)=0$ but $\dim\_F(C+ tC)=1$. This shows that the opposite inequality fails in a rather dramatic function: it does not even hold typically in the sense of Marstrand's Theorem.


Morally speaking, there is no reason why $$\dim\_F(E\_1+E\_2)=\min(1,\dim\_F(E\_1)+\dim\_F(E\_2))$$ should hold in general. Leaving Hausdorff dimension considerations aside, if $E\_1$ or $E\_2$ are not Salem, this tells us that there are some resonances in the construction of the sets at a set of frequencies (possibly very sparse). These special frequencies will in general be lost in the sum $E\_1+E\_2$ (unless $E\_1$ and $E\_2$ also resonate to each other in some strong form), so one would expect that $\dim\_F(E\_1+E\_2) > \dim\_F(E\_1)+\dim\_F(E\_2)$. However, I suspect it is not trivial at all to give specific examples where $\dim\_H(E\_1+E\_2)<1$ (because proving Salemness or even some good decay of Fourier coefficients is usually hard).


On the other hand, equality $\dim\_F(E\_1+E\_2)=\min(1,\dim\_F(E\_1)+\dim\_F(E\_2))$ can certainly hold. Indeed, it is easy to see this is always the case when $E\_1$ and $E\_2$ are Salem and additionally one of them has coinciding Hausdorff and upper box-counting dimension (which is the case for all known constructions of Salem sets). Indeed, denoting upper box-counting dimension by $\dim\_B$, it is well known that $\dim\_H(A+B)\le \dim\_H(A)+\dim\_B(B)$, so in this case $\dim\_H(E\_1+E\_2)\le \dim\_H(E\_1)+\dim\_H(E\_2)$, and it follows from salem-ness and $\dim\_F\le \dim\_H$ that
$$
\dim\_F(E\_1+E\_2)\le \dim\_F(E\_1)+\dim\_F(E\_2).
$$


