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 $\sigma(B(x,r)) \leq r^{s}$. Answering a question of the first author, we show that there exists an exponent $p = p(s,t) \geq 1$ such that $$\|\hat{\sigma}\|_{L^{p}(B(R))} \leq C_{s,t}R^{(2 - t)/p}, \qquad R \geq 1.$$ Moreover, when $s \geq 2/3$ and $t \in [0,s + 1)$, the previous inequality is true for $p \geq 6$. 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 $n \geq 1$ is an integer, then $$ \dim_{\mathrm{H}}(nK) \geq \min\{3s - s \cdot 2^{-(n - 2)},s + 1\}.$$