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.

中文翻译：

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质数域中乘性子群和区间的指数和的新估计

令 ${\mathcal H}$ 是 $\mathbb{F}_p^*$ 阶 $H>p^{1/4}$ 的乘法子群。我们证明 $$ \max_{(a,p)=1}\left|\sum_{x\in {\mathcal H}} {\mathbf{\,e}}_p(ax)\right| \le H^{1-31/2880+o(1)}, $$ where ${\mathbf{\,e}}_p(z) = \exp(2 \pi iz/p)$，提高了Bourgain 和 Garaev (2009) 的结果。我们还获得了乘积 $nx$ 与 $x \in {\mathcal H}$ 和 $n \in {\mathcal N}$ 的双指数和的新估计值，用于连续整数的短间隔 ${\mathcal N}$ .