Related papers: Diophantine approximation with smooth numbers

Let $\mathfrak{p}=(\mathfrak{p}_1,...,\mathfrak{p}_r)$ be a system of $r$ polynomials with integer coefficients of degree $d$ in $n$ variables $\mathbf{x}=(x_1,...,x_n)$. For a given $r$-tuple of integers, say $\mathbf{s}$, a general local…

For two matrices $A$ and $B$, and large $n$, we show that most products of $n$ factors of $e^{A/n}$ and $n$ factors of $e^{B/n}$ are close to $e^{A + B}$. This extends the Lie-Trotter formula. The elementary proof is based on the relation…

We obtain some new inequalities between the ordinary and the uniform Diophantine exponents for simultaneous Diophantine approximation to four real numbers.

A symmetrized lattice of $2n$ points in terms of an irrational real number $\alpha$ is considered in the unit square, as in the theorem of Davenport. If $\alpha$ is a quadratic irrational, the square of the $L^2$ discrepancy is found to be…

For $\lambda \in (1/2, 1)$ and $\alpha$, we consider sets of numbers $x$ such that for infinitely many $n$, $x$ is $2^{-\alpha n}$-close to some $\sum_{i=1}^n \omega_i \lambda^i$, where $\omega_i \in \{0,1\}$. These sets are in Falconer's…

We first recall the connection, going back to A. Thue, between rational approximation to algebraic numbers and integer solutions of some Diophantine equations. Next we recall the equivalence between several finiteness results on various…

The sequence of 1/2-discrepancy sums of $\{x + i \theta \bmod 1\}$ is realized through a sequence of substitutions on an alphabet of three symbols; particular attention is paid to $x=0$. The first application is to show that any asymptotic…

Using a method of H. Davenport and W. M. Schmidt, we show that, for each positive integer n, the ratio 2/n is the optimal exponent of simultaneous approximation to real irrational numbers 1) by all conjugates of algebraic numbers of degree…

We solve Diophantine equations of the type $ a \, (x^3 \!+ \! y^3 \!+ \! z^3 ) = (x \! + \! y \! + \! z)^3$, where $x,y,z$ are integer variables, and the coefficient $a\neq 0$ is rational. We show that there are infinite families of such…

A theorem of Hoischen states that given a positive continuous function $\varepsilon:\mathbb{R}\to\mathbb{R}$, an integer $n\geq 0$, and a closed discrete set $E\subseteq\mathbb{R}$, any $C^n$ function $f:\mathbb{R}\to\mathbb{R}$ can be…

Differentiable real function reproducing primes up to a given number and having a differentiable inverse function is constructed. This inverse function is compared with the Riemann-Von Mangoldt exact expression for the number of primes not…

We present a new, dynamical way to study powers (that is, repetitions) in Sturmian words based on results from Diophantine approximation theory. As a result, we provide an alternative and shorter proof of a result by Damanik and Lenz…

Let $\zeta$ be a real transcendental number. We introduce a new method to find upper bounds for the classical exponent $\widehat{w}_{n}(\zeta)$ concerning uniform polynomial approximation. Our method is based on the parametric geometry of…

We refine a result of W.P. Li and Wang on the values of the form $ \lambda_1p_1 + \lambda_2p_2^{2} + \lambda_3p_3^{2} + \mu_1 2^{m_1} +...+ \mu_s 2^{m_s}, $ where $p_1,p_2,p_3$ are prime numbers, $m_1,..., m_s$ are positive integers,…

Let $F_n$ denote the distribution function of the normalized sum $Z_n = (X_1 + \dots + X_n)/\sigma\sqrt{n}$ of i.i.d. random variables with finite fourth absolute moment. In this paper, polynomial rates of convergence of $F_n$ to the normal…

Let $\Omega(n)$ denote the number of prime factors of $n$. We show that for any bounded $f\colon\mathbb{N}\to\mathbb{C}$ one has \[ \frac{1}{N}\sum_{n=1}^N\, f(\Omega(n)+1)=\frac{1}{N}\sum_{n=1}^N\, f(\Omega(n))+\mathrm{o}_{N\to\infty}(1).…

In this article, for a large class of rational self-similar IFS's wich contains the middle-third Cantor set, we compute the Hausdorff dimension of elements a self-similar set that are $\psi$-approximable by rational belonging to this set…

Let E_n={x_i=1, x_i+x_j=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}}. For a positive integer n, let f(n) denote the greatest finite total number of solutions of a subsystem of E_n in integers x_1,...,x_n. We prove: (1) the function f is…

Counting the number of prime numbers up to a certain natural number and describing the asymptotic behavior of such a counting function has been studied by famous mathematicians like Gauss, Legendre, Dirichlet, and Euler. The prime number…

In this paper we prove inequalities for multiplicative analogues of Diophantine exponents, similar to the ones known in the classical case. Particularly, we show that a matrix is badly approximable if and only if its transpose is badly…