On the size of the maximum of incomplete Kloosterman sums

Let $t:{\mathbb F_p} \to C$ be a complex valued function on ${\mathbb F_p}$ . A classical problem in analytic number theory is bounding the maximum $$M(t): = \mathop {\max }\limits_{0 \le H < p} \left| {{1 \over {\sqrt p }}\sum\limits_{0 \le n < H} {t(n)} } \right|$$ of the absolute value of the incomplete sums $(1/\sqrt p )\sum\nolimits_{0 \le n < H} {t(n)} $ . In this very general context one of the most important results is the Pólya–Vinogradov bound $$M(t) \le {\left\| {\hat t} \right\|_\infty }\log 3p,$$ where $\hat t:{\mathbb F_p} \to \mathbb C$ is the normalized Fourier transform of t. In this paper we provide a lower bound for certain incomplete Kloosterman sums, namely we prove that for any $\varepsilon > 0$ there exists a large subset of $a \in \mathbb F_p^ \times $ such that for $${\rm{k}}{1_{a,1,p}}:x \mapsto e((ax + \bar x)/p)$$ we have $$M({\rm{k}}{1_{a,1,p}}) \ge \left( {{{1 - \varepsilon } \over {\sqrt 2 \pi }} + o(1)} \right)\log \log p,$$ as $p \to \infty $ . Finally, we prove a result on the growth of the moments of ${\{ M({\rm{k}}{1_{a,1,p}})\} _{a \in \mathbb F_p^ \times }}$ .