On the size of the maximum of incomplete Kloosterman sums

<jats:p>Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S030500412100030X_inline1. png" /> <jats:tex-math>$t:{\mathbb F_p} \to C$</jats:tex-math> </jats:alternatives> </jats:inline-formula> be a complex valued function on <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S030500412100030X_inline2. png" /> <jats:tex-math>${\mathbb F_p}$</jats:tex-math> </jats:alternatives> </jats:inline-formula>. A classical problem in analytic number theory is bounding the maximum <jats:disp-formula id="S030500412100030X_udisp1"> <jats:alternatives> <jats:graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S030500412100030X_eqnu1. png" /> <jats:tex-math> $$M(t): = \mathop {\max }\limits_{0 \le H < p} \left| {{1 \over {\sqrt p }}\sum\limits_{0 \le n < H} {t(n)} } \right|$$ </jats:tex-math> </jats:alternatives> </jats:disp-formula> of the absolute value of the incomplete sums <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S030500412100030X_inline3. png" /> <jats:tex-math>$(1/\sqrt p )\sum\nolimits_{0 \le n < H} {t(n)} $</jats:tex-math> </jats:alternatives> </jats:inline-formula>. In this very general context one of the most important results is the Pólya–Vinogradov bound <jats:disp-formula id="S030500412100030X_udisp2"> <jats:alternatives> <jats:graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S030500412100030X_eqnu2. png" /> <jats:tex-math> $$M(t) \le {\left\| {\hat t} \right\|_\infty }\log 3p,$$ </jats:tex-math> </jats:alternatives> </jats:disp-formula> where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S030500412100030X_inline4. png" /> <jats:tex-math>$\hat t:{\mathbb F_p} \to \mathbb C$</jats:tex-math> </jats:alternatives> </jats:inline-formula> is the normalized Fourier transform of <jats:italic>t</jats:italic>. In this paper we provide a lower bound for certain incomplete Kloosterman sums, namely we prove that for any <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S030500412100030X_inline5. png" /> <jats:tex-math>$\varepsilon > 0$</jats:tex-math> </jats:alternatives> </jats:inline-formula> there exists a large subset of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S030500412100030X_inline6. png" /> <jats:tex-math>$a \in \mathbb F_p^ \times $</jats:tex-math> </jats:alternatives> </jats:inline-formula> such that for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S030500412100030X_inline7. png" /> <jats:tex-math>$${\rm{k}}{1_{a,1,p}}:x \mapsto e((ax + \bar x)/p)$$</jats:tex-math> </jats:alternatives> </jats:inline-formula> we have <jats:disp-formula id="S030500412100030X_udisp3"> <jats:alternatives> <jats:graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S030500412100030X_eqnu3. png" /> <jats:tex-math> $$M({\rm{k}}{1_{a,1,p}}) \ge \left( {{{1 - \varepsilon } \over {\sqrt 2 \pi }} + o(1)} \right)\log \log p,$$ </jats:tex-math> </jats:alternatives> </jats:disp-formula> as <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S030500412100030X_inline8. png" /> <jats:tex-math>$p \to \infty $</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Finally, we prove a result on the growth of the moments of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S030500412100030X_inline9. png" /> <jats:tex-math>${\{ M({\rm{k}}{1_{a,1,p}})\} _{a \in \mathbb F_p^ \times }}$</jats:tex-math> </jats:alternatives> </jats:inline-formula>. </jats:p>