An improved lower bound for a problem of Littlewood on the zeros of cosine polynomials
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866917886943035392 |
|---|---|
| author | Bedert, Benjamin |
| author_facet | Bedert, Benjamin |
| contents | Let $Z(N)$ denote the minimum number of zeros in $[0,2π]$ that a cosine polynomial of the form $$f_A(t)=\sum_{n\in A}\cos nt$$ can have when $A$ is a finite set of non-negative integers of size $|A|=N$. It is an old problem of Littlewood to determine $Z(N)$. In this paper, we obtain the lower bound $Z(N)\geqslant (\log\log N)^{(1+o(1))}$ which exponentially improves on the previous best bounds of the form $Z(N)\geqslant (\log\log\log N)^c$ due to Erdélyi and Sahasrabudhe. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_16075 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An improved lower bound for a problem of Littlewood on the zeros of cosine polynomials Bedert, Benjamin Classical Analysis and ODEs Combinatorics Number Theory 26C10, 11C08, 30C15 Let $Z(N)$ denote the minimum number of zeros in $[0,2π]$ that a cosine polynomial of the form $$f_A(t)=\sum_{n\in A}\cos nt$$ can have when $A$ is a finite set of non-negative integers of size $|A|=N$. It is an old problem of Littlewood to determine $Z(N)$. In this paper, we obtain the lower bound $Z(N)\geqslant (\log\log N)^{(1+o(1))}$ which exponentially improves on the previous best bounds of the form $Z(N)\geqslant (\log\log\log N)^c$ due to Erdélyi and Sahasrabudhe. |
| title | An improved lower bound for a problem of Littlewood on the zeros of cosine polynomials |
| topic | Classical Analysis and ODEs Combinatorics Number Theory 26C10, 11C08, 30C15 |
| url | https://arxiv.org/abs/2407.16075 |