Large Salem Sets Avoiding Nonlinear Configurations
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866911372587040768 |
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| author | Denson, Jacob |
| author_facet | Denson, Jacob |
| contents | We construct large Salem sets avoiding patterns, complementing previous constructions of pattern avoiding sets with large Hausdorff dimension. For a (possibly uncountable) family of uniformly Lipschitz functions $\{ f_i : (\mathbb{T}^d)^{n-2} \to \mathbb{T}^d \}$, we obtain a Salem subset of $\mathbb{T}^d$ with dimension $d/(n-1)$ avoiding nontrivial solutions to the equation $x_n - x_{n-1} = f_i(x_1,\dots,x_{n-2})$. For a countable family of smooth functions $\{ f_i : (\mathbb{T}^d)^{n-1} \to \mathbb{T}^d \}$ satisfying a modest geometric condition, we obtain a Salem subset of $\mathbb{T}^d$ with dimension $d/(n-3/4)$ avoiding nontrivial solutions to the equation $x_n = f(x_1,\dots,x_{n-1})$. For a set $Z \subset \mathbb{T}^{dn}$ which is the countable union of a family of sets, each with lower Minkowski dimension $s$, we obtain a Salem subset of $\mathbb{T}^d$ of dimension $(dn - s)/(n - 1/2)$ whose Cartesian product does not intersect $Z$ except at points with non-distinct coordinates. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_09592 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Large Salem Sets Avoiding Nonlinear Configurations Denson, Jacob Classical Analysis and ODEs 42A32 (Primary) 42A38 (Secondary) We construct large Salem sets avoiding patterns, complementing previous constructions of pattern avoiding sets with large Hausdorff dimension. For a (possibly uncountable) family of uniformly Lipschitz functions $\{ f_i : (\mathbb{T}^d)^{n-2} \to \mathbb{T}^d \}$, we obtain a Salem subset of $\mathbb{T}^d$ with dimension $d/(n-1)$ avoiding nontrivial solutions to the equation $x_n - x_{n-1} = f_i(x_1,\dots,x_{n-2})$. For a countable family of smooth functions $\{ f_i : (\mathbb{T}^d)^{n-1} \to \mathbb{T}^d \}$ satisfying a modest geometric condition, we obtain a Salem subset of $\mathbb{T}^d$ with dimension $d/(n-3/4)$ avoiding nontrivial solutions to the equation $x_n = f(x_1,\dots,x_{n-1})$. For a set $Z \subset \mathbb{T}^{dn}$ which is the countable union of a family of sets, each with lower Minkowski dimension $s$, we obtain a Salem subset of $\mathbb{T}^d$ of dimension $(dn - s)/(n - 1/2)$ whose Cartesian product does not intersect $Z$ except at points with non-distinct coordinates. |
| title | Large Salem Sets Avoiding Nonlinear Configurations |
| topic | Classical Analysis and ODEs 42A32 (Primary) 42A38 (Secondary) |
| url | https://arxiv.org/abs/2110.09592 |