0 there exists a large subset of a ∈ F×p such that for kl a,1,p : x → e((ax+x) / p) we have M(kla,1,p) ≥ (1−ε/√2π + o(1)) log log p, as p→∞. Finally, we prove a result on the growth of the moments of {M (kla,1,p)}a∈F×p. 2020 Mathematics Subject Classification: 11L03, 11T23 (Primary); 14F20, 60F10 (Secondary).","lang":"eng"}],"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","short":"CC BY (4.0)"},"intvolume":" 172","has_accepted_license":"1","date_created":"2021-05-02T22:01:29Z","article_type":"original","volume":172,"oa_version":"Published Version","title":"On the size of the maximum of incomplete Kloosterman sums","scopus_import":"1","day":"01","author":[{"last_name":"Bonolis","id":"6A459894-5FDD-11E9-AF35-BB24E6697425","full_name":"Bonolis, Dante","first_name":"Dante"}],"date_published":"2022-05-01T00:00:00Z","acknowledgement":"I am most thankful to my advisor, Emmanuel Kowalski, for suggesting this problem and for his guidance during these years. I also would like to thank Youness Lamzouri for informing me about his work on sum of incomplete Birch sums and Tal Horesh for her suggestions on a previous version of the paper. Finally, I am very grateful to the anonymous referee for their careful reading of the manuscript and their valuable comments.","publication":"Mathematical Proceedings of the Cambridge Philosophical Society","status":"public","external_id":{"isi":["000784421500001"],"arxiv":["1811.10563"]},"isi":1,"year":"2022","quality_controlled":"1","ddc":["510"],"page":"563 - 590","type":"journal_article","_id":"9364","date_updated":"2023-08-02T06:47:48Z","publisher":"Cambridge University Press","article_processing_charge":"Yes (via OA deal)","doi":"10.1017/S030500412100030X"},{"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","citation":{"ama":"Autissier P, Bonolis D, Lamzouri Y. The distribution of the maximum of partial sums of Kloosterman sums and other trace functions. Compositio Mathematica. 2021;157(7):1610-1651. doi:10.1112/s0010437x21007351","short":"P. Autissier, D. Bonolis, Y. Lamzouri, Compositio Mathematica 157 (2021) 1610–1651.","ieee":"P. Autissier, D. Bonolis, and Y. Lamzouri, “The distribution of the maximum of partial sums of Kloosterman sums and other trace functions,” Compositio Mathematica, vol. 157, no. 7. Cambridge University Press, pp. 1610–1651, 2021.","ista":"Autissier P, Bonolis D, Lamzouri Y. 2021. The distribution of the maximum of partial sums of Kloosterman sums and other trace functions. Compositio Mathematica. 157(7), 1610–1651.","chicago":"Autissier, Pascal, Dante Bonolis, and Youness Lamzouri. “The Distribution of the Maximum of Partial Sums of Kloosterman Sums and Other Trace Functions.” Compositio Mathematica. Cambridge University Press, 2021. https://doi.org/10.1112/s0010437x21007351.","mla":"Autissier, Pascal, et al. “The Distribution of the Maximum of Partial Sums of Kloosterman Sums and Other Trace Functions.” Compositio Mathematica, vol. 157, no. 7, Cambridge University Press, 2021, pp. 1610–51, doi:10.1112/s0010437x21007351.","apa":"Autissier, P., Bonolis, D., & Lamzouri, Y. (2021). The distribution of the maximum of partial sums of Kloosterman sums and other trace functions. Compositio Mathematica. Cambridge University Press. https://doi.org/10.1112/s0010437x21007351"},"issue":"7","language":[{"iso":"eng"}],"oa":1,"department":[{"_id":"TiBr"}],"arxiv":1,"month":"06","publication_identifier":{"issn":["0010-437X"],"eissn":["1570-5846"]},"publication_status":"published","intvolume":" 157","abstract":[{"lang":"eng","text":"In this paper, we investigate the distribution of the maximum of partial sums of families of m -periodic complex-valued functions satisfying certain conditions. We obtain precise uniform estimates for the distribution function of this maximum in a near-optimal range. Our results apply to partial sums of Kloosterman sums and other families of ℓ -adic trace functions, and are as strong as those obtained by Bober, Goldmakher, Granville and Koukoulopoulos for character sums. In particular, we improve on the recent work of the third author for Birch sums. However, unlike character sums, we are able to construct families of m -periodic complex-valued functions which satisfy our conditions, but for which the Pólya–Vinogradov inequality is sharp."}],"date_created":"2022-02-01T08:10:43Z","article_type":"original","volume":157,"title":"The distribution of the maximum of partial sums of Kloosterman sums and other trace functions","oa_version":"Preprint","day":"28","author":[{"first_name":"Pascal","last_name":"Autissier","full_name":"Autissier, Pascal"},{"first_name":"Dante","id":"6A459894-5FDD-11E9-AF35-BB24E6697425","full_name":"Bonolis, Dante","last_name":"Bonolis"},{"first_name":"Youness","last_name":"Lamzouri","full_name":"Lamzouri, Youness"}],"date_published":"2021-06-28T00:00:00Z","acknowledgement":"We would like to thank the anonymous referees for carefully reading the paper and for their remarks and suggestions.","publication":"Compositio Mathematica","status":"public","keyword":["Algebra and Number Theory"],"external_id":{"isi":["000667289300001"],"arxiv":["1909.03266"]},"isi":1,"year":"2021","quality_controlled":"1","main_file_link":[{"url":"https://arxiv.org/abs/1909.03266","open_access":"1"}],"page":"1610-1651","type":"journal_article","_id":"10711","date_updated":"2023-08-17T06:59:16Z","publisher":"Cambridge University Press","article_processing_charge":"No","doi":"10.1112/s0010437x21007351"}]