Diophantine approximations with a constant right-hand side of inequalities on short intervals. 1

The problem of finding the Lebesgue measure 𝛍 of the set B1 of the coverings of the solutions of the inequality, ⎸Px⎹ <Q−w, w>n , Q ∈ N and Q >1, in integer polynomials P (x) of degree, which doesn’t exceed n and the height H (P) ≤ Q , is one of the main problems in the metric theory of the Diophantine approximation. We have obtained a new bound 𝛍B1 <c(n)Q−w+n, n<w<n+1, that is the most powerful to date. Even an ineffective version of this bound allowed V. G. Sprindzuk to solve Mahler’s famous problem.

On real algebraic numbers in which the derivative of their minimal polynomial is small

Algebraic numbers are the roots of integer polynomials. Each algebraic number α is characterized by its minimal polynomial Pα that is a polynomial of minimal positive degree with integer coprime coefficients, α being its root. The degree of α is the degree of this polynomial, and the height of α is the maximum of the absolute values of the coefficients of this polynomial. In this paper we consider the distribution of algebraic numbers α whose degree is fixed and height bounded by a growing parameter Q, and the minimal polynomial Pα is such that the absolute value of its derivative P'α (α) is bounded by a given parameter X. We show that if this bounding parameter X is from a certain range, then as Q → +∞ these algebraic numbers are distributed uniformly in the segment [-1+√2/3.1-√2/3]

Universal portfolios generated by rational functions

The f -divergence of Csiszar is defined for a non-negative convex function on the positive axis. A pseudo f -divergence can be defined for a convex function not satisfying the usual requirements. A rational function where both the numerator and the denominator are non-integer polynomials will be used to generate universal portfolios. Five stock-price data sets from the local stock exchange are selected for the empirical study. Empirical results are obtained by running the generated portfolios on these data sets. The empirical results demonstrate that it is possible for the investors to increase their wealth by using the portfolios in investment.

The Solvability of Polynomial Pell’s Equation

This article attempts to describe the continued fraction expansion of ÖD viewed as a Laurent series x-1. As the behavior of the continued fraction expansion of ÖD is related to the solvability of the polynomial Pell’s equation p2-Dq2=1 where D=f2+2g is monic quadratic polynomial with deg g<deg f and the solutions p, q must be integer polynomials. It gives a non-trivial solution if and only if the continued fraction expansion of ÖD is periodic.

ON SPRINDŽUK’S CLASSIFICATION OF -ADIC NUMBERS

We carry Sprindžuk’s classification of the complex numbers to the field $\mathbb{Q}_{p}$ of $p$ -adic numbers. We establish several estimates for the $p$ -adic distance between $p$ -adic roots of integer polynomials, which we apply to show that almost all $p$ -adic numbers, with respect to the Haar measure, are $p$ -adic $\tilde{S}$ -numbers of order 1.

An effective estimate for the measure of the set of p-adic numbers with a given order of approximation

In this paper, we establish a rate of convergence to zero of the measure of the set [Formula: see text] for which the inequality [Formula: see text] for [Formula: see text] has a solution in the integer polynomials [Formula: see text] of degree [Formula: see text] and height bounded by [Formula: see text].

Natural boundaries for Euler products of Igusa zeta functions of elliptic curves

We study the analytic behavior of adelic versions of Igusa integrals given by integer polynomials defining elliptic curves. By applying results on the meromorphic continuation of symmetric power L-functions and the Sato–Tate conjectures, we prove that these global Igusa zeta functions have some meromorphic continuation until a natural boundary beyond which no continuation is possible.