Level sets of prevalent Weierstrass functions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909887260262400 |
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| author | Buczolich, Zoltán Käenmäki, Antti Maga, Balázs |
| author_facet | Buczolich, Zoltán Käenmäki, Antti Maga, Balázs |
| contents | The $α$-Weierstrass function is defined as $W_g^{α,b}(x) = \sum_{k=0}^{\infty} b^{-αk} g(b^k x)$, where $g$ is a Lipschitz function on the unit circle. For a prevalent $α$-Weierstrass function, we prove that the upper Minkowski dimension of every level set is at most $1-α$, and the Hausdorff dimension of almost every level set equals $1-α$ with respect to its occupation measure. We further demonstrate that the occupation measure of a prevalent $α$-Weierstrass function is absolutely continuous with respect to the Lebesgue measure. Consequently, the result on the Hausdorff dimension of level sets applies to a set of level sets with positive Lebesgue measure. A central tool in our analysis is the Weierstrass embedding. For a sufficiently large dimension $d$, we construct Lipschitz functions $g_0, g_1, \ldots, g_{d-1}$ such that the mapping $x \mapsto \big(W_{g_0}^{α,b}(x), W_{g_1}^{α,b}(x), \ldots, W_{g_{d-1}}^{α,b}(x)\big)$ is $α$-bi-Hölder. We also prove that such an embedding requires at least $1/α$ coordinate functions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_15591 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Level sets of prevalent Weierstrass functions Buczolich, Zoltán Käenmäki, Antti Maga, Balázs Classical Analysis and ODEs Primary 28A78, Secondary 26E15, 26A16, 28A50, 46E35, 60G17 The $α$-Weierstrass function is defined as $W_g^{α,b}(x) = \sum_{k=0}^{\infty} b^{-αk} g(b^k x)$, where $g$ is a Lipschitz function on the unit circle. For a prevalent $α$-Weierstrass function, we prove that the upper Minkowski dimension of every level set is at most $1-α$, and the Hausdorff dimension of almost every level set equals $1-α$ with respect to its occupation measure. We further demonstrate that the occupation measure of a prevalent $α$-Weierstrass function is absolutely continuous with respect to the Lebesgue measure. Consequently, the result on the Hausdorff dimension of level sets applies to a set of level sets with positive Lebesgue measure. A central tool in our analysis is the Weierstrass embedding. For a sufficiently large dimension $d$, we construct Lipschitz functions $g_0, g_1, \ldots, g_{d-1}$ such that the mapping $x \mapsto \big(W_{g_0}^{α,b}(x), W_{g_1}^{α,b}(x), \ldots, W_{g_{d-1}}^{α,b}(x)\big)$ is $α$-bi-Hölder. We also prove that such an embedding requires at least $1/α$ coordinate functions. |
| title | Level sets of prevalent Weierstrass functions |
| topic | Classical Analysis and ODEs Primary 28A78, Secondary 26E15, 26A16, 28A50, 46E35, 60G17 |
| url | https://arxiv.org/abs/2507.15591 |