$L^2$-Flattening of Self-similar Measures on Non-degenerate Curves

Let $\mu$ be a non-atomic self-similar measure on $\mathbb{R}$, and let $\nu$ be its pushforward to a non-degenerate curve in ${\mathbb{R}}^d,d\ge 1$. We show that for every $\epsilon >0$, there is $p>1$, so that ${\parallel \widehat{\nu }\parallel }_{L^p(B(R))}^p=O_{\epsilon }(R^{\epsilon })$ for all $R>1$, where $B(R)$ is the $R$-ball about the origin. As a corollary, we show that convolution with $\nu$ quantitatively improves $L^2$-dimension.