chebЧебышевский сборникChebyshevskii Sbornik2226-8383Tula State Lev Tolstoy Pedagogical University10.22405/2226-8383-2015-16-3-124-146cheb-156Research ArticleСтатьиArticleЭКСТРЕМАЛЬНЫЕ ФОРМЫ И ЖЕСТКОСТЬ В АРИФМЕТИЧЕСКОЙ ГЕОМЕТРИИ И В ДИНАМИКЕEXTREMAL FORMS AND RIGIDITY IN ARITHMETIC GEOMETRY AND IN DYNAMICSГлазуновН. М.GlazunovN. M.glanm@yahoo.comNational Aviation UniversityUkraine201503072016163124146Copyright © Глазунов Н.М., 20162016Глазунов Н.М.Glazunov N.M.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://www.chebsbornik.ru/jour/article/view/156С

С. С. Рышков в своих работах исследовал экстремальные формы и экстремальные решетки. Экстремальные формы и экстремальные решетки связаны с жесткими (в смысле М. Громова и других) математическими объектами. В своих работах, а также в работах с коллегами С. С. Рышков пришел и к другим жестким объектам. Жесткие и мягкие задачи, методы и результаты проявляются уже при исследовании классических проблем теории чисел. Остановимся очень кратко на интерпретации с точки зрения жестких и мягких методов бинар- ной и тернарной проблем Гольдбаха, проблем гольдбахова типа и методов их исследования. Так как в бинарной (соответственно, тернарной) пробле- мах Гольдбаха в их современной постановке речь идет о равенствах типа 2n = p1 + p2 (соответственно 2n + 1 = p1 + p2 + p3), где n — натуральное число, большее 1 (соответственно n больше 2), p1, p2, p3 — простые числа, то в своей постановке это жесткие проблемы; результаты их исследования также являются жесткими. Однако методы их исследования включают как жесткие методы — точная формула метода Харди — Литтлвуда — Рамануджана (Х-Л-Р), получаемая методами комплексного анализа, так и сочетание жестких и мягких (soft) методов исследования главного члена в форме Х-Л-Р и оста- точного члена методом тригонометрических сумм Виноградова. Ряд задач аналитической теории чисел допускают динамическую ин- терпретацию. Отметим в связи с этим, что на связи методов аналитической теории чисел и теории динамических систем обращал внимание и развивал такие аналогии А. Г. Постников. Целью предлагаемой работы не является исчерпывающее введение в жесткость в арифметике и в динамике. Скорее мы сделали попытку представить элементарные методы, результаты и некоторые основные идеи в этой области, вместе с обзором ряда новых результатов. Мы не даем ис- черпывающего обзора возможных тем, а также не входим в детали дока- зательств. После представления элементарного теоретико-числового, алгебраического и алгебро-геометрического введения в жесткие неархимедовы пространства на основе локальных одномерных полных регулярных колец, де- ревьев и формальных схем по И. Р. Шафаревичу, Ж.-П. Серру, Дж. Тэйту, Д. Мамфорду, мы даем обзор некоторых новых результатов и методов в направлении жесткости. Изложение включает (но не исчерпывает) результаты и методы H. Furstenberg, G. A. Margulis, G. D. Mostow, R. Zimmer, J. Bourgain, A. Furman, A. Lindenstrauss, S. Mozes, J. James, T. Koberda, K. Lindsey, C. Silva, P. Speh, A. Ioana, K. Kedlaya, J. Tuitman, и других. Я признателен В. М. Бухштаберу за полезные замечания в процессе обсуждения моего доклада. Я благодарю рецензента за замечания относительно содержания и стиля изложения и за предложения по улучшению. Особая признательность Н. М. Добровольскому за помощь и поддержку в процессе подготовки статьи к печати.

 

 

Ryshkov S

Ryshkov S. S. in his papers has investigated extremal forms and extremal lattices. Extremal forms and lattices are connected with hard or rigid (by M. Gromov and other) objects in mathematics. In their work with colleagues S. S. Ryshkov came also to the other hard (or rigid) objects, for instance, to rigidly connected chain. Rigid and soft methods and results already evident in the study of the classical problems in number theory. Let us dwell briefly on the interpretation in terms of hard and soft methods of binary and ternary Goldbach problems. Since the binary (respectively ternary) Goldbach problems in their present formulation there are about equalities of the type 2n = p1 + p2 (respectively 2n+1 = p1+p2+p3), where n is a natural number greater than 1 (respectively n is a natural number greater than 2), p1, p2, p3 prime numbers, then these are hard (rigid) problems; the results of their studies are also hard. However, the methods of their study include both rigid methods — the exact formula of the method of Hardy — Littlewood — Ramanujan and a combination of hard and soft methods under the investigation by the Vinogradov‘s method of trigonometric sums. A number of problems of analytic number theory allow dynamic interpretation. We note in this regard that on connection of methods of analytic number theory and the theory of dynamical systems paid attention and has developed such analogies A. G. Postnikov. The purpose of the paper is not to provide any sort of comprehensive introduction to rigidity in arithmetic and dynamics. Rather, we attempt to convey elementary methods, results and some main ideas of the theory, with a survey of some new results. We do not explore an exhaustive list of possible topics, nor do we go into details in proofs. After giving an elementary number theoretic, algebraic and algebraic geometry introduction to rigid non-Archimedean spaces in the framework of local one dimensional complete regular rings, modules over rings, trees and formal schemes follow to I. R. Shafarevich, J.-P. Serre, J. Tate, D. Mumford, we review some novel results and methods on rigidity. These include (but not exhaust) methods and results by H. Furstenberg, G. A. Margulis, G. D. Mostow, R. Zimmer, J. Bourgain, A. Furman, A. Lindenstrauss, S. Mozes, J. James, T. Koberda, K. Lindsey, C. Silva, P. Speh, A. Ioana, K. Kedlaya, J. Tuitman, and other. I am grateful to V. M. Buchstaber for helpful comments during the discussion of my talk. I thank the anonymous referee for his comments relatively the content and style of presentation and for providing suggestions for improvements. Special thanks to N. M. Dobrovolskii for help and support under the preparation of the article for publication.

 

 

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The authors declare that there are no conflicts of interest present.