Abstract
This survey contains enlarged version of a mini-course which was read by the author in November 2015 during “Chinese - Russian workshop of exponential sums and sumsets”. This workshop was organized by professors Chaohua Jia (Institute of Mathematics, Academia Sinica) and Ke Gong (Henan University) in Academy of Mathematics and System Science, CAS (Beijing). The author is warmly grateful to them for the support and hospitality. The survey contains the Introduction, three parts and Conclusion. The basic definitions and results concerning the complete Kloosterman sums are given in the Introduction. The method of estimating of incomplete Kloosterman sums to moduli equal to the raising power of a fixed prime is described in the first part. This method is based on one idea of A. G. Postnikov which reduces the estimate of such sums to the estimate of the exponential sums with polynomial by I. M. Vinogradov’s mean value theorem. A. A. Karatsuba’s method of estimating of incomplete sums to an arbitrary moduli is described in the second part. This method is based on a very precise estimate of the number of solutions of one symmetric congruence involving inverse residues to a given modulus. This estimate plays the same role in thie problems under considering as Vinogradov’s mean value theorem in the estimating of corresponding exponential sums. The method of J. Bourgain and M. Z. Garaev is described in the third part. This method is based on very deep “sum-product estimate” and on the improvement of A. A. Karatsuba’s bound for the number of solutions of symmetric congruence. The Conclusion contains a series of recent results concerning the estimates of short Kloosterman sums.
Highlights
Настоящий обзор представляет собой развёрнутое содержание мини -курса, прочитанного автором в ноябре 2015 г. во время “Китайско-Российского симпозиума по тригонометрическим суммам и суммам множеств”
This survey contains enlarged version of a mini-course which was read by the author in November 2015 during “Chinese - Russian workshop of exponential sums and sumsets”
This workshop was organized by professors Chaohua Jia (Institute of Mathematics, Academia Sinica) and Ke Gong (Henan University) in Academy of Mathematics and System Science, CAS (Beijing)
Summary
В случае, когда модуль q — простое число, наилучшей для Wq(χ; N, M ) (в смысле длины промежутка суммирования) остаётся оценка Д. В случае простого модуля q наилучшей на сегодняшний день является оценка V ≪ N q−cε, полученная А. В то же время в случае нечётного простого p и q = pn та сумма допускает нетривиальную оценку уже при N ec(log q)2/3 Применяя к неполным суммам Клоостермана по модулю q = pn технику всех перечисленных работ, можно получать их нетривиальные оценки в случаях, когда длина N суммы очень мала по сравнению с модулем. Ниже мы получим ряд оценок неполных сумм Клоостермана и их аналогов для случая, когда модуль q = pn является растущей степенью фиксированного нечётного простого числа p.
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