Related papers: On localization in Kronecker's diophantine theorem

We introduce diophantine approximation groups and their associated Kronecker foliations, using them to provide new algebraic and geometric characterizations of $K$-linear and algebraic dependence. As a consequence we find reformulations --…

We study additive properties of consecutive prime numbers and the primality of the sums they generate. For a given prime number $p_n$, we consider the sums \[ S_k(p_n) = p_n + p_{n+1} + \cdots + p_{n+k-1}, \] where $k \ge 3$ is an odd…

We study the problem of Diophantine approximation on lines in R^2 with prime numerator and denominator.

In the present paper, we have developed a method for solving \textit{diophantine inequalities} using their relationship with the \textit{difference between consecutive primes}. Using this approach we have been able to prove some theorems,…

The distribution of lattice points with relatively $r$-prime is related to problems in the Number Theory such as the Extended Lindel\"{o}f Hypothesis and the Gauss Circle Problem. It is known that Sittinger's result is improved on the…

We prove explicit versions of the Kronecker-Weyl theorems, both in a discrete and a continuous settings, without any linear independence hypothesis. As an application, we propose an alternative approach to problems concerning asymptotic…

We study the problem of Diophantine approximation on lines in $\mathbb{R}^d$ under certain primality restrictions.

This work lies across three areas (in the title) of investigation that are by themselves of independent interest. A problem that arose in quantum computing led us to a link that tied these areas together. This link consists of a single…

The goal of the present paper is to present a method of proving of Diophantine inequalities with primes through the use of auxiliary inequalities and available evaluations of the difference between consecutive primes. We study the Legendre…

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…

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…

Let $K$ be a number field and $\ell \geq 5$ a prime number. Mazur and Rubin introduced the notion of diophantine stability for a variety $X_{/K}$ at a prime $\ell$. We show that there is a positive density set of elliptic curves…

For a number field K and a finite abelian group G, we determine the probabilities of various local completions of a random G-extension of K when extensions are ordered by conductor. In particular, for a fixed prime p of K, we determine the…

For a positive proportion of primes $p$ and $q$, we prove that $\mathbb{Z}$ is Diophantine in the ring of integers of $\mathbb{Q}(\sqrt[3]{p},\sqrt{-q})$. This provides a new and explicit infinite family of number fields $K$ such that…

The recent negative answer to Hilbert's tenth problem over rings of integers relies on a theorem that for every extension of number fields $L/K$, if there is an abelian variety $A$ over $K$ such that $0 < \operatorname{rank} A(K) =…

For an integer $k\geq 2$, let $(L_{n}^{(k)})_{n}$ be the $k-$generalized Lucas sequence which starts with $0,\ldots,0,2,1$ ($k$ terms) and each term afterwards is the sum of the $k$ preceding terms. In this paper, we find all the integers…

Hance and Hossenfelder recently claim that the extensive experimental confirmations of Bell's Theorem do not in fact demonstrate that nature is nonlocal, but merely that nature can be local only if the distant detector settings in a…

Let $d\ge 2$ and $n\ge d$ with $(d,n)\notin \{(2,2),(3,3)\}$. We consider homogeneous Diophantine equations of degree $d$ in $n+1$ variables and whether they have solutions in the primes. In particular, we show that a certain local-global…

Let $\ell$ and $p$ be (not necessarily distinct) prime numbers and $F$ be a global function field of characteristic $\ell$ with field of constants $\kappa$. Assume that there exists a prime $P_\infty$ of $F$ which has degree $1$, and let…

In this extended abstract we deal with the relations between the numerical/diophantine approximation and the symbolic/algebraic geometry approachs to solving of multivariate diophentine polynomial systems, obtaining several consecuences…