  Let $0 \leq s \leq 1$, and let $\mathbb{P} := \{(t,t^{2}) \in \mathbb{R}^{2}
: t \in [-1,1]\}$. If $K \subset \mathbb{P}$ is a closed set with
$\dim_{\mathrm{H}} K = s$, it is not hard to see that $\dim_{\mathrm{H}} (K +
K) \geq 2s$. The main corollary of the paper states that if $0 < s < 1$, then
adding $K$ once more makes the sum slightly larger: $$\dim_{\mathrm{H}} (K + K
+ K) \geq 2s + \epsilon, $$ where $\epsilon = \epsilon(s) > 0$. This
information is deduced from an $L^{6}$ bound for the Fourier transforms of
Frostman measures on $\mathbb{P}$. If $0 < s < 1$, and $\mu$ is a Borel measure
on $\mathbb{P}$ satisfying $\mu(B(x,r)) \leq r^{s}$ for all $x \in \mathbb{P}$
and $r > 0$, then there exists $\epsilon = \epsilon(s) > 0$ such that $$
\|\hat{\mu}\|_{L^{6}(B(R))}^{6} \leq R^{2 - (2s + \epsilon)} $$ for all
sufficiently large $R \geq 1$. The proof is based on a reduction to a
$\delta$-discretised point-circle incidence problem, and eventually to the
$(s,2s)$-Furstenberg set problem.
