Approximation of analytic functions by polynomials with integer coefficients

For some classes of analytic functions we estimate the rate of approximation by polynomials with integer coefficients.Translated fromMatematicheskie Zametki, Vol. 59, No. 2, pp. 182–188, February, 1996.     

Limit theorems for polynomial-type distributions

Distributions are found of independent nonnegative integer valued random variables under linear constraints. Limit theorems for these distributions are proved. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipolez, pp. 136–148, Perm, 1993.     

Bounded three-dimensional Fano varieties of integer index

The cube of the anticanonical class of a three-dimensional Fano variety with canonical singularities and integer Fano index is effectively bounded. Translated fromMatematicheskie Zametki, Vol. 66, No. 3, pp. 445–451, September, 1999.     

Torsion free Abelian groups of finite rank and their direct decompositions

This article presents a new method, based on the theory of integer representations, for investigating torsion free abelian groups of finite rank. In particular, it is proved that the direct decompositions of such groups are in bijective correspondence with the decompositions of vectors of some cone in an integer lattice into the sum of vectors in the lattice.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 175, pp. 135–153, 1989.     

Representation of large numbers by binary quadratic forms

Sufficient conditions of representability of a given integer by a fixed binary quadratic form are presented. It is shown that almost all integers satisfy these conditions. Bibliography: 2 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 60–64.     

Symmetric multiplicative statistics

A family of multiplicative statistics on integer partitions that are symmetric under the conjunction of Young diagrams is constructed. It is shown that all symmetric multiplicative statistics lie in this family. A limit shape of Young diagrams under these statistics is found.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 240, 1997, pp. 280–289.     

Nonhomogeneous waring equations

It is proved that for an arbitrary positive integer k the equation n=x²+y²+z³+u³+v⁴+w¹⁴+t^4k+1 has a positive integer solution for all sufficiently large n. Bibliography: 6 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 65–68.     

On Gauss-Kuz’min statistics for finite continued fractions

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. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 6, pp. 195–208, 2005.     

Polynomials with high multiplicity at unity and Tarry’s problem

The existence of a polynomial with integer coefficients of relatively small length and sufficiently large multiplicity at unity is established. The proof of the corresponding statement is based on an estimate of the number of solutions of Tarry’s system. Translated fromMatematicheskie Zametki, Vol. 65, No. 6, pp. 810–815, June, 1999.     

On Galois spectra of polynomials with integral parameters

We prove that there exists a polynomial F(x, t) with rational coefficients, whose degree with respect to x is equal to 4, such that for every integer a, the Galois group of the decomposition field of the polynomial F(x, a) is not the dihedral group, but any other transitive subgroup of the group S₄ can be represented as the Galois group of the decomposition field of the polynomial F(x, a) for a certain integer a. Bibliography: 1 title. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 321, 2005, pp. 275–280.     

On the Coincidence of Limit Shapes for Integer Partitions and Compositions,and a Slicing of Young Diagrams

We consider the slicing of Young diagrams into slices associated with summands that have equal multiplicities. It is shown that for the uniform measure on all partitions of an integer n, as well as for the uniform measure on partitions of an integer n into m summands with m ∼ An^α, α ≤ 1/2, all slices after rescaling concentrate around their limit shapes. The similar problem is solved for compositions of an integer n into m summands. These results explain why the limit shapes of partitions and compositions coincide in the case α < 1/2. Bibliography: 10 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 307, 2004, pp. 266–280.     

Differentiation of analytic functions of sectorial operators in noncommuting directions

We find expressions and estimates for integer powers of the differentiation operator in noncommuting directions for analytic functions of sectorial operators on a complex Banach space. We study the exponential of such a differentiation operator and give an application to the theory of perturbations of evolution equations.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 4, 1997, pp. 70–74.     

Lower bounds for linear forms in values of certain hypergeometric functions

By using Padé approximations of the first kind, a lower bound for the modulus of a linear form with integer coefficients in the values of certain hypergeometric functions at a rational point are obtained. This estimate depends on all the coefficients of the linear form. Translated fromMatematicheskie Zametki, Vol. 67, No. 3, pp. 441–451, March, 2000.     

Remark on asymptotic distribution of integer points on a large three-dimensional sphere

New estimates that are uniform in weight are obtained for the Fourier coefficients of parabolic forms of half-integer weight. The results are applied to the well-known problem of the distribution of integer points on three-dimensional ellipsoids.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 185, pp. 22–28, 1990.     

The adic realization of the Morse transformation and the extension of its action to the solenoid

We consider the adic realization of the Morse transformation on the additive group of integer dyadic numbers. We discuss the arithmetic properties of this action. Then we extend this action to an action of the group of rational dyadic numbers on the solenoid. Bibliography: 14 titles. To the memory of Alexander Livshits Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 360, 2008, pp. 70–90.     

Lower bound for a linear form

By using Padé approximations of the first kind, we obtain a lower bound for the absolute value of a linear form with integer coefficients from the values of polylogarithmic functions at rational points. This estimate takes into account the growth of all coefficients of the linear form. Translated fromMatematicheskie Zametki, Vol. 66, No. 4, pp. 617–623, October, 1999.     

Expansion of vectors by powers of a matrix

In this paper, we investigate the problem of expansion of any d-dimensional vector in powers of a dilation matrix M, where a dilation matrix is an integer matrix of size d × d with all modules of its eigenvalues more than one. We consider this expansion as a multidimensional system of numeration, where we take the matrix as the base of the system of numeration and a special set of vectors as the set of digits. We give a constructive method of expansion of integer vectors in powers of a dilation matrix and prove the existence of expansion for any real vector. Bibliography: 4 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 355, 2008, pp. 199–218.     

Strengthening theC t -closing lemma for dynamical systems and foliations on the torus

We prove a strengthenedC ^r -closing lemma (r≥1) for wandering chain recurrent trajectories of flows without equilibrium states on the two-dimensional torus and for wandering chain recurrent orbits of a diffeomorphism of the circle. The strengthenedC ^r -closing lemma (r≥1) is proved for a special class of infinitely smooth actions of the integer lattice ℤ ^k on the circle. The result is applied to foliations of codimension one with trivial holonomy group on the three-dimensional torus. Translated fromMatematicheskie Zametki, Vol. 61, No. 3, pp. 323–331, March, 1997. Translated by S. K. Lando     

Comparison of various generalizations of continued fractions

We use the Euler, Jacobi, Poincaré, and Brun matrix algorithms as well as two new algorithms to evaluate the continued fraction expansions of two vectorsL related to two Davenport cubic formsg ₁ andg ₂. The Klein polyhedra ofg ₁ andg ₂ were calculated in another paper. Here the integer convergentsP _k given by the cited algorithms are considered with respect to the Klein polyhedra. We also study the periods of these expansions. It turns out that only the Jacobi and Bryuno algorithms can be regarded as satisfactory. Translated fromMatematicheskie Zametki, Vol. 61, No. 3, pp. 339–348, March, 1997. Translated by V. E. Nazaikinskii     

Exponents of elements of a normal basis of the ideal of algebraic functions on a three-sheeted Riemannian surface

On a three-sheeted Riemannian surfaceR of genus given by an irreducible algebraic equation, we construct normal bases of the ideal of algebraic functions that are multiples of some integer divisors. A method for constructing such normal bases was given in [V. E. Kruglov,Dokl. Akad. Nauk SSSR,321, No. 1, 11–13 (1991)]. The relations obtained for the exponents of the constructed elements enable one to determine the number of solutions of the Riemann problem for any integer divisor and to find partial indices in the problems of factorization of matrices of permutation type.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 8, pp. 1029–1037, August, 1995.