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Статистика траекторий частиц в однородной задаче Синая для двумерной решетки
Функциональный анализ и его приложения. 2008. Т. 42. № 3. С. 10–22.
Быковский В. А., Ustinov A.
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 function for the free path length and for the sighting parameter (the distance from the trajectory to the integer point in question) does not depend on the particle escape direction (the property of isotropy).
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
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
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
Ustinov A., Квант 2010 № 2 С. 32–33
The article discusses applications of continued fractions. ...
Added: October 12, 2025
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
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
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
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
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
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
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
Ustinov A., Быковский В. А., Известия РАН. Серия математическая 2009 Т. 73 № 4 С. 17–36
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 ...
Added: October 10, 2025
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
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
Ustinov A., Известия РАН. Серия математическая 2010 Т. 74 № 5 С. 145–170
We prove the existence of the limit density distribution for normalized Frobenius numbers with three arguments. The density is found explicitly. ...
Added: October 9, 2025