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...

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Veröffentlicht in:arXiv.org 2020-03
Hauptverfasser: Daniel di Benedetto, Garaev, Moubariz Z, García, Víctor C, González-Sánchez, Diego, Shparlinski, Igor E, Trujillo, Carlos A
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Sprache:eng
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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