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arXiv · 2507.07321

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

Amir Algom·Osama Khalil
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Abstract

Let $\mu$ be a non-atomic self-similar measure on $\mathbb{R}$, and let $\nu$ be its pushforward to a non-degenerate curve in $\mathbb{R}^d, d\geq 1$. We show that for every $\epsilon>0$, there is $p>1$, so that $\left \lVert \hat{\nu} \right \rVert_{L^p(B(R))}^p = O_\epsilon(R^\epsilon)$ 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.

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Amir Algom, Osama Khalil. 2025-07-09. $L^2$-Flattening of Self-similar Measures on Non-degenerate Curves. https://arxiv.org/abs/2507.07321

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