Related papers: Diophantine approximation with primes from short i…

Diophantine approximation is traditionally the study of how well real numbers are approximated by rationals. We propose a model for studying Diophantine approximation in an arbitrary totally bounded metric space where the rationals are…

Our goal is to finally settle the persistent problem in Diophantine Approximation of finding best linear approximates. Classical results from the theory of continued fractions provide the solution for the special homogeneous case in the…

A number alpha in R is diophantine if it is not well approximable by rationals, i.e. for some C, nu>0 and any relatively prime p, q in Z we have |alpha q -p|>C q^{-1-\vu}. It is well-known and easy to prove that almost every alpha in R is…

We study the problem of best approximations of a vector $\alpha\in{\mathbb R}^n$ by rational vectors of a lattice $\Lambda\subset {\mathbb R}^n$ whose common denominator is bounded. To this end we introduce successive minima for a periodic…

We explicitly construct a diffeomorphic pair (p(x),p^{-1}(x)) in terms of an appropriate quadric spline interpolating the prime series. These continuously differentiable functions are the smooth analogs of the prime series and the prime…

Answering an informal question of K. Park, we show that by fixing some irrational alpha to have a particular standard continued fraction expansion, we may force the associated discrepancy sequences for all x in [0,1), which track the…

Diophantine approximation explores how well irrational numbers can be approximated by rationals, with foundational results by Dirichlet, Hurwitz, and Liouville culminating in Roth's theorem. Schmidt's subspace theorem extends Roth's results…

In 2004, J.C. Tong found bounds for the approximation quality of a regular continued fraction convergent of a rational number, expressed in bounds for both the previous and next approximation. We sharpen his results with a geometric method…

For any given real number $\alpha$ with bounded partial quotients, we construct explicitly continuum many real numbers $\beta$ with bounded partial quotients for which the pair $(\alpha, \beta)$ satisfies a strong form of the Littlewood…

We give criteria of the solvability of the diophantine equation $p=x^2+ny^2$ over some imaginary quadratic fields where $p$ is a prime element. The criteria becomes quite simple in special cases.

Let $\lfloor t\rfloor$ denote the integer part of $t\in\mathbb{R}$ and $\|x\|$ the distance from $x$ to the nearest integer. Suppose that $1/2<\gamma_2<\gamma_1<1$ are two fixed constants. In this paper, it is proved that, whenever $\alpha$…

We discuss the problem of finding optimal exponents in Diophantine estimates involving one real number and, in some cases where such an exponent is known, present some properties of the corresponding extremal numbers.

A well known theorem of Lagrange states that the simple continued fraction of a real number $\alpha$ is periodic if and only if $\alpha$ is a quadratic irrational. We examine non-periodic and non-simple continued fractions formed by two…

In this paper we present a new approach to prove effective results in Diophantine approximation. We then use it to prove an effective theorem on the simultaneous approximation of two algebraic numbers satisfying an algebraic equation with…

We study the distribution of the values of the form $\lambda_1 p_1 + \lambda_2 p_2 + \lambda_3 p_3^k$, where $\lambda_1$, $\lambda_2$ and $\lambda_3$ are non-zero real number not all of the same sign, with $\lambda_1 / \lambda_2$…

Given $n\in N$ and $x,\gamma\in R$, let \begin{equation*} ||\gamma-nx||^\prime=\min\{|\gamma-nx+m|:m\in Z, \gcd (n,m)=1\}, \end{equation*} %where $(n,m)$ is the largest common divisor of $n$ and $m$. Two conjectures in the coprime…

In this paper we consider the problem of counting algebraic numbers $\alpha$ of fixed degree $n$ and bounded height $Q$ such that the derivative of the minimal polynomial $P_{\alpha}(x)$ of $\alpha$ is bounded, $|P_{\alpha}'(\alpha)| <…

We prove that if $\lambda_1$, $\lambda_2$, $\lambda_3$ and $\lambda_4$ are non-zero real numbers, not all of the same sign, $\lambda_1 / \lambda_2$ is irrational, and $\varpi$ is any real number then, for any $\eps > 0$ the inequality $…

We prove a Khintchine result for convergence of a multiplicative Diophantine set with restricted denominators on an arbitrary non-degenerate line. Specifically, given sequences of real numbers $\{a_n\}_{n\in\mathbb{N}},\,…

Fundamental questions in Diophantine approximation are related to the Hausdorff dimension of sets of the form $\{x\in \mathbb{R}: \delta_x = \delta\}$, where $\delta \geq 1$ and $\delta_x$ is the Diophantine approximation rate of an…