Ustinov Alexey Vladimirovich

Head of Department of Theoretical and Applied Mathematics

Editorial Duties:
Uniform Distribution Theory
Mathematical Enlightenment
Far Eastern Mathematical Journal

E-mail address:
Phone: +7(4212) 45-69-12
Fax: +7(4212) 32-46-76

Main publications

Number Theory

  1. On the number of summands in the asymptotic formula for the number of solutions to Waring equation. Mathematical Notes, 1998, 64:2, 240-250. (Link to the Journal, pdf 109 kb).
  2. On summation and interpolation formulas. Proceedings of the IV International Conference «Modern Problems of Number Theory and its Applications» (Russian) (Tula, 2001), Chebyshevskii Sb., 2001, 1, 52--71. (Link to the Journal, pdf 378 kb, in Russian).
  3. A Discrete Analog of Euler's Summation Formula. Mathematical Notes, 2002, 71:6, 851-856. (Link to the Journal, pdf 109 kb).
  4. On a Diophantine inequality. Chebyshevskii Sb., 2002, 3:1(3), 103--119. (Link to the Journal).
  5. On a generalization of Stirling numbers. Dedicated to the 85th birthday of Nikolai Mikhailovich Korobov (Russian). Chebyshevskii Sb., 2002, 3:2(4), 107--122. (Link to the Journal, pdf 332 kb, in Russian, ps 468 kb, in Russian).
  6. The scientific creativity of Nikolai Mikhailovich Korobov. Dedicated to the 85th birthday of Nikolai Mikhailovich Korobov (Russian). Chebyshevskii Sb., 2002, 3:2(4), 8--15. (with V.A. Bykovskii and N.M. Dobrovolskii). (Link to the Journal, pdf 118 kb, in Russian).
  7. On some cubic equations. Fundam. Prikl. Mat., 2002, 8:1, 263-271. (Link to the Journal, pdf 153 kb, in Russian).
  8. An elementary approach to the computation of $\zeta(2n)$. Dedicated to the 75th birthday of Aleksandr Vasil'evich Malyshev (Russian). Chebyshevskii Sb., 2003, 4:3(7), 106--110. (Link to the Journal, pdf 299 kb, in Russian).
  9. Korobov polynomials and umbral analysis. Chebyshevskii Sb., 2003, 4:4(8), 137--152. (Link to the Journal, pdf 371 kb, in Russian).
  10. A Discrete Analog of the Poisson Summation Formula. Mathematical Notes, 2003, 73:1, 97-102. (Link to the Journal, pdf 112 kb).
  11. Applications of special polynomials. Appendix in book of N.M. Korobov Applications of number theory to numerical analysis. 2004, Moscow, MCCME, 11 pp. (Russian).
  12. Korobov polynomials. Proceedings of the VIII International Conference «Discrete mathematics and its applications», 2004, Moscow, 233-236 (Russian). (pdf 190 kb, in Russian).
  13. On the statistical properties of finite continued fractions. Journal of Mathematical Sciences (New York), 2006, 137:2, 4722-4738. (Link to the Journal, pdf 469 kb).
  14. On Gauss-Kuz'min statistics for finite continued fractions. Journal of Mathematical Sciences (New York), 2007, 146:2, 5771-5781. (Link to the Journal, pdf 137 kb).
  15. A Short Proof of Euler's Identity for Continuants. Mathematical Notes, 2006, 79:1, 146-147. (Link to the Journal, pdf 77 kb).
  16. Calculation of the variance in a problem in the theory of continued fractions. Sbornik: Mathematics, 2007, 198:6, 887-907. (Link to the Journal, pdf 312 kb).
  17. On average number of steps in Euclidean algorithm. Materials of the IX regional contest of young scientists, 2007. (pdf 353 kb, in Russian).
  18. The Statistics of Particle Trajectories in the Homogeneous Sinai Problem for a Two-Dimensional Lattice. Functional Analysis and Its Applications, 2008, 42:3, 169-179. (with V.A. Bykovskii). (Link to the Journal, pdf 177 kb).
  19. Asymptotic behaviour of the first and second moments for the number of steps in the Euclidean algorithm. Izvestiya: Mathematics, 2008, 72:5, 1023-1059. (Link to the Journal, pdf 394 kb).
  20. On the number of solutions of the congruence $xy=l (mod q)$. St. Petersburg Mathematical Journal, 2009, 20:5, 813-836. (Link to the Journal, pdf 268 kb).
  21. Applications of Kloosterman sums in some problems of metrical and analytical number theory. D.Sc. (Habilitation) thesis, Khabarovsk, 2008. (pdf 1.36 Mb, in Russian).
  22. The Mean Number of Steps in the Euclidean Algorithm with Least Absolute-Value Remainders. Mathematical Notes, 2009, 85:1, 142-145. (Link to the Journal, pdf 442 kb).
  23. On the statistical properties of elements of continued fractions. Dokl. Akad. Nauk, 2009, 424:4, 459-461. (pdf 117 kb).
  24. The statistics of particle trajectories in the inhomogeneous Sinai problem for a two-dimensional lattice. Izvestiya: Mathematics, 2009, 73:4, 669-688. (with V.A. Bykovskii). (Link to the Journal, pdf 271 kb).
  25. Limiting distribution of Frobenius numbers for $n=3$ (with V. Shchur and Ya. Sinai). Journal of Number Theory, 2009, 129:11, 2778-2789. (pdf 205 kb).
  26. On the distribution of integer points. Far Eastern Mathematical Journal, 2009, 9:1-2, 176-181. (pdf 468 kb, in Russian).
  27. The solution of Arnold's problem on the weak asymptotics of Frobenius numbers with three arguments. Sbornik: Mathematics, 2009, 200:4, 597-627. (Link to the Journal, pdf 398 kb).
  28. The mean number of steps in the Euclidean algorithm with odd partial quotients. Mathematical Notes, 2010, 88:4, 574-584. (pdf 532 kb).
  29. On the distribution of Frobenius numbers with three arguments. Izvestiya:Mathematics, 2010, 74:5, 1023-1049. (pdf 587 kb).
  30. Applications of Kloosterman Sums in Arithmetic and Geometry. (In Russian). LAMBERT Academic Publishing, 2011. (Link to the Publishing House).
  31. On Gauss - Kuz'min statistics in short intervals. FEMJ, 11:1 (2011), 93-98. (pdf 215 kb, in Russian).
  32. On Voronoi's cylindric minima theorem. FEMJ, 11:2 (2011), 213-221. (pdf 287 kb, in Russian).
  33. Geometric proof of Rödseth's formula for Frobenius numbers. Proceedings of the Steklov Institute of Mathematics, 276 (2012), 275-282. (Link to the Journal, pdf 1.28 Mb).
  34. Minimal Vector Systems in Three-Dimensional Lattices and an Analogue of Vahlen's Theorem for Three-Dimensional Minkowski Continued Fractions. Sovremennye Problemy Matematiki, 2012, Vol. 16, pp. 103–128. (Link to the Journal, pdf 255 kb).
  35. Spin chains and Arnold's problem on the Gauss-Kuz'min statistics for quadratic irrationals. Sbornik: Mathematics, 204:5 (2013), 762-779. (Link to the Journal, pdf 373 kb).
  36. On the Three-Dimensional Vahlen Theorem. Mat. Zametki, 95:1 (2014), 154-156. (Link to the Journal, pdf 453 kb).
  37. Calculation of a Gauss sum via the discrete Fourier transform. FEMJ, 14:1 (2014), 90-95. (pdf 115 kb, in Russian).
  38. On continued fractions of equal length. FEMJ, 14:1 (2014), 96-99. (pdf 96 kb, in Russian).
  39. Geometry of Minkowski-Voronoi tessellations of the plane (with O. Karpenkov), arXiv:1407.0135. (Link to the Journal, pdf 275 kb).
  40. On the distribution of points on the modular hyperbola. FEMJ, 14:2 (2014), 141–155. (with V.I. Bernik). (Link to the Journal, pdf 209 kb, in Russian).
  41. The distribution of solutions of a determinantal equation. Sbornik: Mathematics, 206:7 (2015), 988–1019. (Link to the Journal).
  42. Coefficient rings of formal group laws. Sbornik: Mathematics, 206:11 (2015), 1524–1563. (with V.M. Buchstaber). (Link to the Journal).
  43. Three-dimensional continued fractions and Kloosterman sums. Russian Mathematical Surveys, 70:3(423) (2015), 483–556. (Link to the Journal).
  44. On a generalization of Kloosterman sums. Mathematical Notes, 97:1 (2015), 147–151. (Link to the Journal).
  45. On points of the modular hyperbola under the graph of a linear function. Mathematical Notes, 97:2 (2015), 284–288. (Link to the Journal).
  46. Distribution of algebraic points in domains of small measure and near the surfaces. Chebyshevskii Sb., 16:3 (2015), 78–94. (with V.I. Bernik, A.G. Gusakova). (Link to the Journal). pdf 341 kb, in Russian).
  47. Supercontinuants. Appendix 2 in the article "Introducing supersymmetric frieze patterns and linear difference operators". Mathematische Zeitschrift, December 2015, Volume 281, Issue 3, pp. 1061-1087. (Link to the Journal, see also).

Elementary Mathematics

  1. Variants of entrance examinations in A.N. Kolmogorov's school in physics and mathematics. 1999, Moscow, 80 pp. (with N.B. Alfutova, T.P. Korneeva and M.V. Smurov).
  2. Variants of entrance examinations in A.N. Kolmogorov's school in physics and mathematics. 2000, Moscow, 80 pp. (with N.B. Alfutova, T.P. Korneeva, M.V. Smurov and V.V. Zagorskii). (djvu 1.37 Mb, in Russian).
  3. Algebra and number theory for mathematical schools. Collected problems. 2002, Moscow, MCCME, 264 pp. (Russian, with N.B Alfutova). (pdf 2 Mb).
  4. Algebra and number theory for mathematical schools. Collected problems. 2005, Moscow, MCCME, 320 pp. (Russian, the second edition, with N.B. Alfutova).
  5. Problems on a Lattices. In the book «Summer school in AESC MSU-2004». 2005, Moscow: URSS, 44-49 (Russian, with O.N. German and V.V. Vavilov). (pdf 214 kb, in Rusian).
  6. Polygons on the lattices. 2006, Moscow, MCCME, 70 pp. (Russian, with V. V. Vavilov). (pdf 725 kb, in Russian).
  7. 18 x 18 problems of PhMS. No. 18. 2006, Moscow, MCCME, 160 pp. (Russian, with N.B. Alfutova and Yu.A. Egorov).
  8. Problems on the squared paper. Kolmogorov's School, 2006, 72 pp. (Russian, with V.V. Vavilov).
  9. Systems of linear equations. In the book Summer school in AESC MSU-2005, 2006, Moscow, MCCME. (pdf 203 kb, in Russian).
  10. Kaleidoscope of problems. In the book Summer school in AESC MSU-2005, 2006, Moscow, MCCME (Russian, with O.N. German and V.V. Vavilov). (pdf 135 kb, in Russian).
  11. Circles on the lattices. Kvant, 2006, No. 6, 10-14 (Russian, with V.V. Vavilov). (pdf 919 kb, in Russian).
  12. Problems on the squared paper-I. Mathematical Education, 2006, No. 4(39). (Russian, with V.V. Vavilov).
  13. Semi-regular polygons on the lattices. Kvant, 2007, No. 6, 13-15 (Russian, with V.V. Vavilov). (pdf 152 kb, in Russian).
  14. Problems on the squared paper-II. Mathematical Education, 2007, No. 2(42). (Russian, with V.V. Vavilov).
  15. Two famous formulae. Kvant, 2008, No. 2, 11-15, 22 (Russian, with V.V. Vavilov). (pdf 344 kb, in Russian).
  16. Integer points in the regions. «Potential», 2008, No. 9, 53. (pdf 128 kb, in Russian).
  17. Two difficult problems on descent method. «Potential», 2009, No. 3, 55-56. (pdf 357 kb, in Russian).
  18. On the Construction of a Triangle from the Feet of Its Angle Bisectors. Forum Geometricorum, 2009, No. 9, 279-280. (pdf 24 kb).
  19. Algebra and number theory for mathematical schools. Collected problems. 2009, Moscow, MCCME (Russian, the third edition, with N.B. Alfutova). (pdf 2 Mb, in Russian).
  20. Continued fractions around us. «Kvant», 2010, No. 2, 32-33. (pdf 1.54 Mb, in Russian).
  21. Resistance between two nodes on a lattice. (with D. Baranov and M. Skopenkov). «Mathematical Enlightenment», third series, volume 15 (2011), 229-230. (pdf 393 kb, in Russian).
  22. Chests and protocols. Cryptography without formulae. First Khabarovsk Congress of Teachers of Mathematics (abstracts), 99-101. Khabarovsk, Pacific National University, 2011. (pdf 94 kb, in Russian).
  23. Random walks and electrical networks. (with M. Skopenkov and V. Smykalov) «Mathematical Enlightenment», third series, volume 16 (2012), 25-47. (pdf 694 kb, in Russian).
  24. Whether it is possible to reconstruct a triangle from the feet of its angle bisectors? «Potential»: Mathematics, Physics and Informatics, No. 10 (2013), 41-50. (pdf 507 kb, in Russian), Preprint (pdf 450 kb, in Russian).
  25. 18 x 18. Entrance examinations of PhMS. 2nd edition, Moscow, MCCME, 2014. (Russian, with N.B. Alfutova and Yu.A. Egorov).
    (It can be bought in e-shop Logobook), Look inside (pdf 132 kb, in Russian).
  26. Across the resistor network. (with M. Skopenkov and A. Pakharev). «Mathematical Enlightenment», third series, volume 18 (2014), 33-65. (pdf 958 kb, in Russian).


  1. Kloosterman Sums and Continued Fractions. (Poster from conference «Analysis of Algorithms-2010»). (pdf 160 kb).
  2. Kloosterman Sums and Continued Fractions. International Conference on Uniform Distribution Theory, July 5 - 9, 2010, Strobl (Austria). (pdf 600 kb)
  3. Random walks through electrical networks. 22nd summer conference International mathematical Tournament of cities . (with M. Skopenkov and D. Baranov). (pdf 465 kb).
  4. Why do we need Gauss-Kuz'min statistics? (a talk given at the conference «Diophantine approximation: state of the art and applications», Minsk, 2011). (pdf 0.97 Mb).
  5. Voronoi - Minkowski 3D continued fractions (slides). (pdf 586 kb, Video).
  6. The Farey Fraction Spin Chain and Gauss—Kuz’min Statistics for Quadratic Irrationals. 3rd International Conference on Uniform Distribution Theory (Smolenice, Slovakia, June 25 - June 29, 2012). (pdf 476 kb).
  7. On statistical properties of 3D Voronoi-Minkowski continued fractions. 4th International Conference on Uniform Distribution Theory (June 30 – July 4, 2014, Czech Republic). (pdf 620 kb).
  8. International conference «Geometry, Topology and Integrability», Mathematical Scientific Program (Skolkovo Institute of Science and Technology, Moscow, 20–25 October, 2014).
    Website of the conference, Abstracts
  9. International conference «Torus Actions in Geometry, Topology, and Applications», Mathematical Scientific Program (Skolkovo Institute of Science and Technology, Moscow, 16-21 February, 2015).
    Website of the conference, All slides, Author's slides
  10. Double Somos-4 "29th Journées Arithmetiques" (July 6-10, 2015, Debrecen, Hungary). (pdf 552 kb).
  11. Distribution of Frobenius Numbers. “International Meeting on Numerical Semigroups with Applications”, Levico Terme (Trento, Italy) on July 4th - 8th, 2016. (pdf 608 kb).