Discrete Fourier restriction via efficient congruencing: basic principles
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2015
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866911935253970944 |
|---|---|
| author | Wooley, Trevor D. |
| author_facet | Wooley, Trevor D. |
| contents | We show that whenever $s>k(k+1)$, then for any complex sequence $(\mathfrak a_n)_{n\in \mathbb Z}$, one has $$\int_{[0,1)^k}\left| \sum_{|n|\le N}\mathfrak a_ne(α_1n+\ldots +α_kn^k) \right|^{2s}\,{\rm d}{\mathbf α}\ll N^{s-k(k+1)/2}\biggl( \sum_{|n|\le N}|\mathfrak a_n|^2\biggr)^s.$$ Bounds for the constant in the associated periodic Strichartz inequality from $L^{2s}$ to $l^2$ of the conjectured order of magnitude follow, and likewise for the constant in the discrete Fourier restriction problem from $l^2$ to $L^{s'}$, where $s'=2s/(2s-1)$. These bounds are obtained by generalising the efficient congruencing method from Vinogradov's mean value theorem to the present setting, introducing tools of wider application into the subject. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1508_05329 |
| institution | arXiv |
| publishDate | 2015 |
| record_format | arxiv |
| spellingShingle | Discrete Fourier restriction via efficient congruencing: basic principles Wooley, Trevor D. Classical Analysis and ODEs Number Theory 42B05, 11L07, 42A16 We show that whenever $s>k(k+1)$, then for any complex sequence $(\mathfrak a_n)_{n\in \mathbb Z}$, one has $$\int_{[0,1)^k}\left| \sum_{|n|\le N}\mathfrak a_ne(α_1n+\ldots +α_kn^k) \right|^{2s}\,{\rm d}{\mathbf α}\ll N^{s-k(k+1)/2}\biggl( \sum_{|n|\le N}|\mathfrak a_n|^2\biggr)^s.$$ Bounds for the constant in the associated periodic Strichartz inequality from $L^{2s}$ to $l^2$ of the conjectured order of magnitude follow, and likewise for the constant in the discrete Fourier restriction problem from $l^2$ to $L^{s'}$, where $s'=2s/(2s-1)$. These bounds are obtained by generalising the efficient congruencing method from Vinogradov's mean value theorem to the present setting, introducing tools of wider application into the subject. |
| title | Discrete Fourier restriction via efficient congruencing: basic principles |
| topic | Classical Analysis and ODEs Number Theory 42B05, 11L07, 42A16 |
| url | https://arxiv.org/abs/1508.05329 |