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 $$ \dim_F(E_1+E_2)\le \dim_F(E_1)+\dim_F(E_2). $$

Regarding the Hausdorff dimension of sumsets, you might want to have a look at an important paper by Peres and Shmerkin:

http://arxiv.org/abs/0705.2628

There are plenty of useful references there, too.