On the local Fourier uniformity problem for small sets
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866914913533820928 |
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| author | Kanigowski, Adam Lemańczyk, Mariusz Richter, Florian Karl Teräväinen, Joni |
| author_facet | Kanigowski, Adam Lemańczyk, Mariusz Richter, Florian Karl Teräväinen, Joni |
| contents | We consider vanishing properties of exponential sums of the Liouville function $λ$ of the form $$ \lim_{H\to\infty}\limsup_{X\to\infty}\frac{1}{\log X}\sum_{m\leq X}\frac{1}{m}\sup_{α\in C}\bigg|\frac{1}{H}\sum_{h\leq H}λ(m+h)e^{2πihα}\bigg|=0, $$ where $C\subset\mathbb{T}$. The case $C=\mathbb{T}$ corresponds to the local $1$-Fourier uniformity conjecture of Tao, a central open problem in the study of multiplicative functions with far-reaching number-theoretic applications. We show that the above holds for any closed set $C\subset\mathbb{T}$ of zero Lebesgue measure. Moreover, we prove that extending this to any set $C$ with non-empty interior is equivalent to the $C=\mathbb{T}$ case, which shows that our results are essentially optimal without resolving the full conjecture. We also consider higher-order variants. We prove that if the linear phase $e^{2πihα}$ is replaced by a polynomial phase $e^{2πih^tα}$ for $t\geq 2$ then the statement remains true for any set $C$ of upper box-counting dimension $<1/t$. The statement also remains true if the supremum over linear phases is replaced with a supremum over all nilsequences coming form a compact countable ergodic subsets of any $t$-step nilpotent Lie group. Furthermore, we discuss the unweighted version of the local $1$-Fourier uniformity problem, showing its validity for a class of ``rigid'' sets (of full Hausdorff dimension) and proving a density result for all closed subsets of zero Lebesgue measure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_05528 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the local Fourier uniformity problem for small sets Kanigowski, Adam Lemańczyk, Mariusz Richter, Florian Karl Teräväinen, Joni Dynamical Systems 37A44, 11N37 We consider vanishing properties of exponential sums of the Liouville function $λ$ of the form $$ \lim_{H\to\infty}\limsup_{X\to\infty}\frac{1}{\log X}\sum_{m\leq X}\frac{1}{m}\sup_{α\in C}\bigg|\frac{1}{H}\sum_{h\leq H}λ(m+h)e^{2πihα}\bigg|=0, $$ where $C\subset\mathbb{T}$. The case $C=\mathbb{T}$ corresponds to the local $1$-Fourier uniformity conjecture of Tao, a central open problem in the study of multiplicative functions with far-reaching number-theoretic applications. We show that the above holds for any closed set $C\subset\mathbb{T}$ of zero Lebesgue measure. Moreover, we prove that extending this to any set $C$ with non-empty interior is equivalent to the $C=\mathbb{T}$ case, which shows that our results are essentially optimal without resolving the full conjecture. We also consider higher-order variants. We prove that if the linear phase $e^{2πihα}$ is replaced by a polynomial phase $e^{2πih^tα}$ for $t\geq 2$ then the statement remains true for any set $C$ of upper box-counting dimension $<1/t$. The statement also remains true if the supremum over linear phases is replaced with a supremum over all nilsequences coming form a compact countable ergodic subsets of any $t$-step nilpotent Lie group. Furthermore, we discuss the unweighted version of the local $1$-Fourier uniformity problem, showing its validity for a class of ``rigid'' sets (of full Hausdorff dimension) and proving a density result for all closed subsets of zero Lebesgue measure. |
| title | On the local Fourier uniformity problem for small sets |
| topic | Dynamical Systems 37A44, 11N37 |
| url | https://arxiv.org/abs/2310.05528 |