On exponential sums over multiplicative subgroups of medium size (Q405956)

Let \(p\) be a prime number, \(\mathbb{F}_p\) be the finite field, \(\mathbb{F}_p^*=\mathbb{F}_p\setminus \{0\}\), and let \(\Gamma\subseteq \mathbb{F}_p^*\) be a multiplicative subgroup. Define NEWLINE\[NEWLINEM(\Gamma)=\max_{\xi\neq 0}|\sum_{x\in \Gamma}e^{\frac{2\pi ix\xi}{p}}|,\quad k\Gamma=\left\{\gamma_1+\cdots+\gamma_k: \gamma_j\in \Gamma, j=1,\cdots,k\right\},\quad S(a)=\sum_{n=1}^{p}e^{\frac{2\pi ian^p}{p^2}}.NEWLINE\]NEWLINE The author proves the following results:NEWLINENEWLINE1) For \( |\Gamma|\leq p^{2/3}\), we have \(M(\Gamma)\ll |\Gamma|^{1/2}p^{1/6}\log^{1/6}|\Gamma|\).NEWLINENEWLINE2) For \( |\Gamma|\gg p^{1/2} \log^{1/3} p\) and \(-1\in \Gamma\), we have \(\mathbb{F}_p^*\subseteq 5\Gamma\).NEWLINENEWLINE3) For \(a\not\equiv 0 \pmod p\), we have \(|S(a)|\ll p^{5/6}\log^{1/6}p\).