On some methods of evaluating irrationality measure of the function arctan 𝑥 values

A new integral construction unifying the idea of symmetry proposed by Salikhov in 2007 and the integral introduced by Markovecchio in 2009 is considered. The application of this construction leads, in particular, to a sharper estimate of the measure of irrationality of the number π/√3.

By using Pade 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.

Linear recurrent sequences and irrationality measures

The article is devoted to the latest results in metric theory of Diophantine approximation. One of the first major result in area of number theory was a theorem by academician Jonas Kubilius. This paper is dedicated to centenary of his birth. Over the last 70 years, the area of Diophantine approximation yielded a number of significant results by great mathematicians, including Fields prize winners Alan Baker and Grigori Margulis. In 1964 academician of the Academy of Sciences of BSSR Vladimir Sprindžuk, who was a pupil of academician J. Kubilius, solved the well-known Mahler’s conjecture on the measure of the set of S-numbers under Mahler’s classification, thus becoming the founder of the Belarusian academic school of number theory in 1962.

On some further one-sided theorems of new type in the theory of diophantine approximations

In the metric theory of Diophantine approximations, one of the main problems leading to exact characteristics in the classifications of Mahler and Koksma is to estimate the Lebesgue measure of the points x ∈ B ⊂ I from the interval I such as the inequality | P (x) | < Q-w, w > n, Q >1 for the polynomials P(x) ∈ Z[x], deg P ≤ n, H(P) ≤Q is satisfied. The methods of obtaining estimates are different at different intervals of w change. In this article, at w > n +1 we get the estimate µ B< c1(n) Q – (w-1/n). The best estimate to date was c2(n) Q –(w- n/n).

The concept of the discriminant of a quadratic polynomial allows for easy extraction of information about its real and complex roots. The discriminant of a polynomial of an arbitrary degree is also an important characteristic of the polynomial, which proves useful in many problems in the theory of Diophantine approximation. In 2023, Belarusian mathematician D. Badziahin solved a problem posed by Davenport in the 1960s concerning the range of values of discriminants in the cubic case. The paper provides a complete solution to the problem of divisibility of discriminants by large powers of prime numbers in the case of cubic polynomials.

We consider sequences of the form $\left(a_{n} \alpha\right)_{n}$ mod 1, where $\alpha\in\left[0,1\right]$ and where $\left(a_{n}\right)_{n}$ is a strictly increasing sequence of positive integers. If the asymptotic distribution of the pair correlations of this sequence follows the Poissonian model for almost all $\alpha$ in the sense of Lebesgue measure, we say that $(a_n)_n$ has the metric pair correlation property. Recent research has revealed a connection between the metric theory of pair correlations of such sequences, and the additive energy of truncations of $(a_n)_{n}$. Bloom, Chow, Gafni and Walker speculated that there might be a convergence/divergence criterion which fully characterises the metric pair correlation property in terms of the additive energy, similar to Khintchine's criterion in the metric theory of Diophantine approximation. In the present paper we give a negative answer to such speculations, by showing that such a criterion does not exist. To this end, we construct a sequence $(a_n)_n$ having large additive energy which, however, maintains the metric pair correlation property.

The golden number \(\omega =\frac{\sqrt{5}-1}{2}\) is not only useful for the golden section method, but it also plays an important role in the theory of diophantine approximation. This inspired us to think about the connection between numerical integration and the golden number. In 1960 we found the very efficient quadrature formula $$\int\limits_{0}^{1}{{}}\int\limits_{0}^{1}{{}}f\left( x,y \right)dxdy\approx \frac{1}{{{F}_{m}}}\sum\limits_{k=1}^{{{F}_{m}}}{{}}f\left( \frac{k}{{{F}_{m}}}\left\{ \frac{{{F}_{m}}-{{1}^{k}}}{{{F}_{m}}}\left. {} \right\} \right. \right)$$ (5.1) that is based on the rational approximation of ω $$\left| \frac{{{F}_{m}}-1}{{{F}_{m}}}-\omega \right|<\frac{1}{\sqrt{5}F_{m}^{2}}$$ and where {ع} denotes the fractional part of ع.

Presented in this paper is a more detailed version of the talk given at the international conference “Recent trends in ergodic theory and Dynamical systems” (University of Vadodara, India, December 26–29, 2012). In 1998 D. Kleinbock and G. Margulis have proved V. Sprindzuk’s hypothesis on extremality of almost all points of nondegenerate manifolds. Their proof was based on the results of S. Dani and G. Margulis in the theory of dynamical systems. In this article, metric theorems on solvability of systems of inequalities involving polynomials and their derivatives are used to obtain results on the distribution of algebraic numbers, as well as discriminants and resultants of integer polynomials.

On a measure of algebraic independence of the values of elliptic functions was published in Approximations Diophantiennes et Nombres Transcendants. Diophantine Approximations and Transcendental Numbers on page 239.

Methods of the theory of transcendental numbers, Diophantine approximations and solutions of Diophantine equations

A symmetric, square-integrable, continuous and positive kernel, considered as a generalization of Bailey’s Poisson kernel, is shown to have the Jacobi polynomials $P_n^{(\alpha ,\beta )} (x)$ as eigenfunctions with eigenvalues expressed as infinite series with arbitrary sequence-coefficients. The consequent bilinear formula, via Mercer’s theorem, includes, as special cases, Bailey’s sum, Bateman’s and Gegenbauer’s degenerate addition formulas for Bessel functions and Feidheim’s projection formula for Jacobi polynomials. In another special case the formula leads to the integrated version of a well-known addition theorem for the sum of a product of three Jacobi polynomials.

Let ξ \xi be a real irrational number. We are interested in sequences of linear forms in 1 and ξ \xi , with integer coefficients, which tend to 0. Does such a sequence exist such that the linear forms are small (with given rate of decrease) and the coefficients have some given rate of growth? If these rates are essentially geometric, a necessary condition for such a sequence to exist is that the linear forms are not too small, a condition which can be expressed precisely using the irrationality exponent of ξ \xi . We prove that this condition is actually sufficient, even for arbitrary rates of growth and decrease. We also make some remarks and ask some questions about multivariate generalizations connected to Fischler-Zudilin’s new proof of Nesterenko’s linear independence criterion.

We will show that this construction is impossible if a straightedge and compass are the only tools which we may use. We precede the proof with a couple of definitions and facts which will be needed. A complex number is algebraic if it is a zero of a polynomial with integer coefficients. Numbers which are not algebraic are called transcendental. The set of all algebraic numbers is a field. This implies, in particular, that for any algebraic number a and any natural number k the number ak iS also algebraic. Recall also that each constructible number, i.e., a number which can be constructed using a straightedge and compass only, is algebraic. A crucial result for us is the following theorem due to Lindemann. If x 7s 0 is an algebraic number, then ex is a transcendental number. We refer to [2, 3] for unexplained terminology and details.