New estimates for exponential sums over multiplicative subgroups and intervals in prime fields
Let \({\mathcal H}\) be a multiplicative subgroup of \(\mathbb{F}_p^*\) of order \(H>p^{1/4}\). We show that $$ \max_{(a,p)=1}\left|\sum_{x\in {\mathcal H}} {\mathbf{\,e}}_p(ax)\right| \le H^{1-31/2880+o(1)}, $$ where \({\mathbf{\,e}}_p(z) = \exp(2 \pi i z/p)\), which improves a result of Bourgai...
Gespeichert in:
Veröffentlicht in: | arXiv.org 2020-03 |
---|---|
Hauptverfasser: | , , , , , |
Format: | Artikel |
Sprache: | eng |
Schlagworte: | |
Online-Zugang: | Volltext |
Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
Zusammenfassung: | Let \({\mathcal H}\) be a multiplicative subgroup of \(\mathbb{F}_p^*\) of order \(H>p^{1/4}\). We show that $$ \max_{(a,p)=1}\left|\sum_{x\in {\mathcal H}} {\mathbf{\,e}}_p(ax)\right| \le H^{1-31/2880+o(1)}, $$ where \({\mathbf{\,e}}_p(z) = \exp(2 \pi i z/p)\), which improves a result of Bourgain and Garaev (2009). We also obtain new estimates for double exponential sums with product \(nx\) with \(x \in {\mathcal H}\) and \(n \in {\mathcal N}\) for a short interval \({\mathcal N}\) of consecutive integers. |
---|---|
ISSN: | 2331-8422 |