• A
  • A
  • A
  • АБВ
  • АБВ
  • АБВ
  • A
  • A
  • A
  • A
  • A
Обычная версия сайта
  • RU
  • EN
  • HSE University
  • Publications
  • Articles
  • Статистика траекторий частиц в неоднородной задаче Синая для двумерной решетки
  • RU
  • EN
Расширенный поиск
Высшая школа экономики
Национальный исследовательский университет
Priority areas
  • business informatics
  • economics
  • engineering science
  • humanitarian
  • IT and mathematics
  • law
  • management
  • mathematics
  • sociology
  • state and public administration
by year
  • 2027
  • 2026
  • 2025
  • 2024
  • 2023
  • 2022
  • 2021
  • 2020
  • 2019
  • 2018
  • 2017
  • 2016
  • 2015
  • 2014
  • 2013
  • 2012
  • 2011
  • 2010
  • 2009
  • 2008
  • 2007
  • 2006
  • 2005
  • 2004
  • 2003
  • 2002
  • 2001
  • 2000
  • 1999
  • 1998
  • 1997
  • 1996
  • 1995
  • 1994
  • 1993
  • 1992
  • 1991
  • 1990
  • 1989
  • 1988
  • 1987
  • 1986
  • 1985
  • 1984
  • 1983
  • 1982
  • 1981
  • 1980
  • 1979
  • 1978
  • 1977
  • 1976
  • 1975
  • 1974
  • 1973
  • 1972
  • 1971
  • 1970
  • 1969
  • 1968
  • 1967
  • 1966
  • 1965
  • 1964
  • 1963
  • 1958
  • More
Subject
News
October 14, 2026
At HSE University–St Petersburg, Experts Discuss How AI Changes Tourist Routes and Cultural Heritage
Neural networks are no longer simple travel assistants; now they decide what to include on the route and what to skip. Artificial intelligence changes how people perceive a place and shapes the image of tourist centres. At the II International Scientific and Practical Conference ’Information Technologies in the Preservation and Promotion of Cultural Heritage,’ researchers discussed who controls algorithmic solutions and in what ways, and whether authentic traditions get lost in them.
October 12, 2026
Road Injuries Leading Cause of Death Among Men Aged 10 to 39
Worldwide, road injuries are the leading cause of death among males aged 10–39. An international team of researchers reached this conclusion as part of the Global Burden of Disease Study 2023, analysing data from 204 countries between 1990 and 2023. Their findings have been published in The Lancet Public Health, with HSE University Professor Vasily Vlassov listed among the collaborators.
October 8, 2026
HSE Experts Take Part in 23rd Annual Meeting of Valdai Discussion Club
The 23rd Annual Meeting of the Valdai Discussion Club was held from September 28 to October 1, 2026 under the theme ‘Responsibility for the Future: Limits of the Possible, or Limitless Possibilities?’ The forum brought together 120 experts from 40 countries, including representatives of China, the United States, India, Brazil, the United Kingdom, Germany, Egypt, Iran, and Japan.

 

Have you spotted a typo?
Highlight it, click Ctrl+Enter and send us a message. Thank you for your help!

Publications
  • Books
  • Articles
  • Chapters of books
  • Working papers
  • Report a publication
  • Research at HSE

?

Статистика траекторий частиц в неоднородной задаче Синая для двумерной решетки

Известия РАН. Серия математическая. 2009. Т. 73. № 4. С. 17–36.
Ustinov A., Быковский В. А.

In connection with the two-dimensional model known as the ‘periodic Lorentz gas’, we study the asymptotic behaviour of statistical characteristics of a free path interval of a point particle before its first occurrence in an h-neighbourhood (a circle of radius h) of a non-zero integer point as h→0 given that the particle starts from the h-neighbourhood of the origin. We evaluate the limit distribution function of the free path length and of the input aimed parameter (the distance from the trajectory to the integer point we are interested in) for a given value of the output aimed parameter. This problem was studied earlier for a particle starting from the origin (the homogeneous case).

Language: Russian
DOI
Text on another site
Keywords: continued fractionsгеометрия чиселцепные дробиGeometry of numbersKloosterman sumsсуммы Клостермана
Similar publications
Hausdorff dimension estimates for Sudler products with positive lower bound
Dmitry Gayfulin, Hauke M., Nonlinearity 2025 Vol. 38 No. 6 Article 065008
Given an irrational number $\alpha$, we study the asymptotic behaviour of the Sudler product denoted by $P_N(\alpha) =\prod_{r=1}^N  2\lvert \sin \pi r \alpha \rvert$. We show that $\liminf_{N \to \infty} P_N(\alpha) >0$ and $\limsup_{N \to \infty} P_N(\alpha)/N < \infty$ whenever the sequence of partial quotients in the continued fraction expansion of $\alpha$ exceeds 3 only finitely ...
Added: March 19, 2026
О равномерных диофантовых экспонентах решеток
German O., Математический сборник 2026 Т. 217 № 2 С. 20–33
The spectrum of values of the weak uniform Diophantine exponents of lattices is studied, and its complete description in the two-dimensional case is given. ...
Added: February 20, 2026
Теоретико-числовые методы в приближенном анализе
Коробов Н. М., МЦНМО, 2004.
Added: October 16, 2025
Приложения сумм Клостермана в арифметике и геометрии
Ustinov A., LAP LAMBERT Academic Publishing, 2011.
The book is devoted to applications of Kloosterman sum estimates in various problems of number theory. ...
Added: October 13, 2025
Цепные дроби вокруг нас
Ustinov A., Квант 2010 № 2 С. 32–33
The article discusses applications of continued fractions. ...
Added: October 12, 2025
О статистических свойствах конечных цепных дробей
Ustinov A., Записки научных семинаров ПОМИ РАН 2005 Т. 322 С. 186–211
The article is devoted to the statistical properties of continued fractions for the numbers a/b, for a and b in the sector a,b⩾1, a^2+b^2⩽R^2. ...
Added: October 11, 2025
О статистиках Гаусса–Кузьмина для конечных цепных дробей
Ustinov A., Фундаментальная и прикладная математика 2005 Т. 11 № 6 С. 195–208
The article is devoted to finite continued fractions for numbers a/b when integer points (a,b) are taken from a dilative region. Properties similar to the Gauss–Kuz'min statistics are proved for these continued fractions. ...
Added: October 11, 2025
Короткое доказательство тождества Эйлера для континуантов
Ustinov A., Математические заметки 2006 Т. 79 № 1 С. 155–156
The article presents a short proof of Euler's identity for continuants ...
Added: October 11, 2025
Вычисление дисперсии в одной задаче из теории цепных дробей
Ustinov A., Математический сборник 2007 Т. 198 № 6 С. 139–158
This paper examines a random variable equal to the number of denominators of convergents not exceeding a given bound. Asymptotic formulas with two significant terms are proved for the mathematical expectation of this variable and its variance. ...
Added: October 11, 2025
Асимптотическое поведение первого и второго моментов для числа шагов в алгоритме Евклида
Ustinov A., Известия РАН. Серия математическая 2008 Т. 72 № 5 С. 189–224
We prove asymptotic formulae with two significant terms for the expectation and variance of the random variable s(c/d) when the variables c and d range over the set 1≤c≤d≤R and R→∞, where s(c,d)=s(c/d) is the number of steps in the Euclidean algorithm applied to the numbers c and d. ...
Added: October 11, 2025
Статистика траекторий частиц в однородной задаче Синая для двумерной решетки
Быковский В. А., Ustinov A., Функциональный анализ и его приложения 2008 Т. 42 № 3 С. 10–22
In this paper, we generalize and refine some results by F. P. Boca, R. N. Gologan, and A. Zaharescu on the asymptotic behavior as h→0 of the statistics of the free path length until the first hit of the h-neighborhood (a disk of radius h) of a nonzero integer for a particle issuing from the origin. The established facts imply that the limit distribution ...
Added: October 11, 2025
О числе решений сравнения xy≡l (mod q) под графиком дважды непрерывно дифференцируемой функции
Ustinov A., Алгебра и анализ 2008 Т. 20 № 5 С. 186–216
A result by V. A.Bykovskiĭ (1981) on the number of solutions of the congruence xy≡l (modq) under the graph of a twice continuously differentiable function is refined. As an application, Porter's result (1975) on the mean number of steps in the Euclid algorithm is sharpened and extended to the case of Gauss–Kuzmin statistics. ...
Added: October 11, 2025
Решение задачи Арнольда о слабой асимптотике для чисел Фробениуса с тремя аргументами
Ustinov A., Математический сборник 2009 Т. 200 № 4 С. 131–160
It is shown that on the average the Frobenius numbers f(a,b,c) behave like 8/π√abc . ...
Added: October 11, 2025
О распределении точек целочисленной решетки
Ustinov A., Дальневосточный математический журнал 2009 Т. 9 № 1-2 С. 176–181
We study distribution of distances from primitive integer points to the origin. ...
Added: October 10, 2025
О статистических свойствах элементов цепных дробей
Ustinov A., Доклады Академии наук 2009 Т. 424 № 4 С. 459–461
The article solves a problem related to the statistical properties of continued fractions that arose during the study of Frobenius numbers with three arguments. ...
Added: October 10, 2025
  • About
  • About
  • Key Figures & Facts
  • Sustainability at HSE University
  • Faculties & Departments
  • International Partnerships
  • Faculty & Staff
  • HSE Buildings
  • HSE University for Persons with Disabilities
  • Public Enquiries
  • Studies
  • Admissions
  • Programme Catalogue
  • Undergraduate
  • Graduate
  • Exchange Programmes
  • Summer University
  • Summer Schools
  • Semester in Moscow
  • Business Internship
  • Research
  • International Laboratories
  • Research Centres
  • Research Projects
  • Monitoring Studies
  • Conferences & Seminars
  • Academic Jobs
  • Yasin (April) International Academic Conference on Economic and Social Development
  • Media & Resources
  • Publications by staff
  • HSE Journals
  • Publishing House
  • iq.hse.ru: commentary by HSE experts
  • Library
  • Economic & Social Data Archive
  • Video
  • HSE Repository of Socio-Economic Information
  • HSE1993–2026
  • Contacts
  • Copyright
  • Privacy Policy
  • Site Map
Edit