Fourier transform of nonlinear images of self-similar measures: quantitative aspects
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866929751660167168 |
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| author | Banaji, Amlan Yu, Han |
| author_facet | Banaji, Amlan Yu, Han |
| contents | This paper relates to the Fourier decay properties of images of self-similar measures $μ$ on $\mathbb{R}^k$ under nonlinear smooth maps $f \colon \mathbb{R}^k \to \mathbb{R}$. For example, we prove that if the linear parts of the similarities defining $μ$ commute and the graph of $f$ has nonvanishing Gaussian curvature, then the Fourier dimension of the image measure is at least $\max\left\{ \frac{2(2κ_2 - k)}{4 + 2κ_* - k} , 0 \right\}$, where $κ_2$ is the lower correlation dimension of $μ$ and $κ_*$ is the Assouad dimension of the support of $μ$. Under some additional assumptions on $μ$, we use recent breakthroughs in the fractal uncertainty principle to obtain further improvements for the decay exponents.
We give several applications to nonlinear arithmetic of self-similar sets $F$ in the line. For example, we prove that if $\dim_{\mathrm H} F > (\sqrt{65} - 5)/4 = 0.765\dots$ then the arithmetic product set $F \cdot F = \{ xy : x,y \in F \}$ has positive Lebesgue measure, while if $\dim_{\mathrm H} F > (-3 + \sqrt{41})/4 = 0.850\dots$ then $F \cdot F \cdot F$ has non-empty interior. One feature of the above results is that they do not require any separation conditions on the self-similar sets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_07508 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fourier transform of nonlinear images of self-similar measures: quantitative aspects Banaji, Amlan Yu, Han Dynamical Systems Classical Analysis and ODEs 42A16 (Primary) 28A80 (Secondary) This paper relates to the Fourier decay properties of images of self-similar measures $μ$ on $\mathbb{R}^k$ under nonlinear smooth maps $f \colon \mathbb{R}^k \to \mathbb{R}$. For example, we prove that if the linear parts of the similarities defining $μ$ commute and the graph of $f$ has nonvanishing Gaussian curvature, then the Fourier dimension of the image measure is at least $\max\left\{ \frac{2(2κ_2 - k)}{4 + 2κ_* - k} , 0 \right\}$, where $κ_2$ is the lower correlation dimension of $μ$ and $κ_*$ is the Assouad dimension of the support of $μ$. Under some additional assumptions on $μ$, we use recent breakthroughs in the fractal uncertainty principle to obtain further improvements for the decay exponents. We give several applications to nonlinear arithmetic of self-similar sets $F$ in the line. For example, we prove that if $\dim_{\mathrm H} F > (\sqrt{65} - 5)/4 = 0.765\dots$ then the arithmetic product set $F \cdot F = \{ xy : x,y \in F \}$ has positive Lebesgue measure, while if $\dim_{\mathrm H} F > (-3 + \sqrt{41})/4 = 0.850\dots$ then $F \cdot F \cdot F$ has non-empty interior. One feature of the above results is that they do not require any separation conditions on the self-similar sets. |
| title | Fourier transform of nonlinear images of self-similar measures: quantitative aspects |
| topic | Dynamical Systems Classical Analysis and ODEs 42A16 (Primary) 28A80 (Secondary) |
| url | https://arxiv.org/abs/2503.07508 |