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О распределении чисел Фробениуса с тремя аргументами
Известия РАН. Серия математическая. 2010. Т. 74. № 5. С. 145–170.
Ustinov A.
We prove the existence of the limit density distribution for normalized Frobenius numbers with three arguments. The density is found explicitly.
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
Shchur V., Sinai Y. G., Ustinov A., Journal of Number Theory 2009 Vol. 129 No. 11 P. 2778–2789
The purpose of this paper is to give a complete derivation of the limiting distribution of large Frobenius numbers outlined in earlier work of J. Bourgain and Ya. Sinai and fill some gaps formulated there as hypotheses. ...
Added: October 12, 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
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