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О распределении точек целочисленной решетки
Дальневосточный математический журнал. 2009. Т. 9. № 1-2. С. 176–181.
Ustinov A.
We study distribution of distances from primitive integer points to the origin.
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
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
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., 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
Ustinov A., Дальневосточный математический журнал 2011 Т. 11 № 1 С. 93–98
The article is devoted to investigation of Gauss — Kuz'min statistics for rational numbers a/b, where b is fixed, 1⩽a⩽b, (a,b)=1. New asymptotic formula for the mean value of Gauss — Kuz'min statistics is proved. It sharpens previous result which is similar to the Porter's theorem. ...
Added: October 9, 2025
Ustinov A., Математический сборник 2013 Т. 204 № 5 С. 143–160
New results related to number theoretic model of spin chains are proved. We solve Arnold's problem on the Gauss-Kuz'min statistics for quadratic irrationals. ...
Added: October 9, 2025
Ustinov A., Берник В. И., Дальневосточный математический журнал 2014 Т. 14 № 2 С. 141–155
It is known that solutions of determinant equation det∣∣axyz∣∣=q are uniformly distributed. The article contains similar results when the variables satisfy additional restrictions (a,x)=1 or (a,x,y,z)=1. ...
Added: October 9, 2025
Ustinov A., Математический сборник 2015 Т. 206 № 7 С. 103–134
In 1964, Linnik and Skubenko established the equidistribution of the integral points on the determinantal surface detX=P, where X is a (3×3) matrix with independent entries and P is an increasing parameter. Their method involved reducing the problem by one dimension (that is, to the determinantal equations with a (2×2) matrix). In this paper a more precise version of the Linnik-Skubenko reduction is proposed. It ...
Added: October 9, 2025
Ustinov A., Успехи математических наук 2015 Т. 70 № 3 С. 107–180
This survey is devoted to results related to metric properties of classical continued fractions and Voronoi–Minkowski three-dimensional continued fractions. The main focus is on applications of analytic methods based on estimates of Kloosterman sums. An apparatus is developed for solving problems about three-dimensional lattices. The approach is based on reduction to the preceding dimension, an ...
Added: October 9, 2025