On Fourier transforms of fractal measures on the parabola

Let $s\in [0,1]$ and $t\in [0,\min \{3s,s+1\})$. Let $\sigma$ be a Borel measure supported on the parabola $\mathbb{P}=\{(x,x^2):x\in [-1,1]\}$ satisfying the $s$-dimensional Frostman condition $\sigma (B(x,r))\le r^s$. Answering a question of the first author, we show that there exists an exponent $p=p(s,t)\ge 1$ such that $$\mathrm{\parallel }\widehat{\sigma }{\mathrm{\parallel }}_{L^p(B(R))}\le C_{s,t}{R}^{(2-t)\mathrm{/}p},R\ge 1.$$ Moreover, when $s\ge 2\mathrm{/}3$ and $t\in [0,s+1)$, the previous inequality is true for $p\ge 6$. We also obtain the following fractal geometric counterpart of the previous results. If $K\subset \mathbb{P}$ is a Borel set with ${\dim }_{\mathrm{H}}K=s\in [0,1]$, and $n\ge 1$ is an integer, then $${\dim }_{\mathrm{H}}(nK)\ge \min \{3s-s\cdot 2^{-(n-2)},s+1\}\mathrm{.}$$