The trigonometric polynomial on sums of two squares, an additive problem and generalisation
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866918149729812480 |
|---|---|
| author | Ramare, Olivier Viswanadham, GK |
| author_facet | Ramare, Olivier Viswanadham, GK |
| contents | Let $B$ be the set of odd integers that are sums of two coprime squares. We prove that the trigonometric polynomial $S(α;N)=\sum_{b\in B,b\leq N} e(bα)$ satisfies \[ \frac{S(α; N)}{N/\sqrt{\log N}}<<_{A,A'} \frac{1}{ϕ(q)} + \sqrt{\frac{q}{N}}(\log N)^{7} +\frac{1}{(\log N)^A} \] for any $A,A'\geq 0$ and when $(a,q)=1$ and $|qα-a|\leq (\log N)^{A'}/N$. We use this estimate together with a variant of the circle method influenced by Green and Tao's Transference Principle to obtain the number of representations of a large enough odd integer $N$ as a sum $b+b_1+b_2$, where $b\in B$ while $b_1$ (resp. $b_2$) belongs to a general subset $B_1$ (resp. $B_2$) of $B$ of relative positive density. We further show that the above bound is effective when $0\leq A<1/2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_23260 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The trigonometric polynomial on sums of two squares, an additive problem and generalisation Ramare, Olivier Viswanadham, GK Number Theory 11L07 (Primary) 11M26, 11N25, 11P55 (Secondary) Let $B$ be the set of odd integers that are sums of two coprime squares. We prove that the trigonometric polynomial $S(α;N)=\sum_{b\in B,b\leq N} e(bα)$ satisfies \[ \frac{S(α; N)}{N/\sqrt{\log N}}<<_{A,A'} \frac{1}{ϕ(q)} + \sqrt{\frac{q}{N}}(\log N)^{7} +\frac{1}{(\log N)^A} \] for any $A,A'\geq 0$ and when $(a,q)=1$ and $|qα-a|\leq (\log N)^{A'}/N$. We use this estimate together with a variant of the circle method influenced by Green and Tao's Transference Principle to obtain the number of representations of a large enough odd integer $N$ as a sum $b+b_1+b_2$, where $b\in B$ while $b_1$ (resp. $b_2$) belongs to a general subset $B_1$ (resp. $B_2$) of $B$ of relative positive density. We further show that the above bound is effective when $0\leq A<1/2$. |
| title | The trigonometric polynomial on sums of two squares, an additive problem and generalisation |
| topic | Number Theory 11L07 (Primary) 11M26, 11N25, 11P55 (Secondary) |
| url | https://arxiv.org/abs/2509.23260 |