Related papers: On localization in Kronecker's diophantine theorem

The overall aim of this note is to initiate a "manifold" theory for metric Diophantine approximation on the limit sets of Kleinian groups. We investigate the notions of singular and extremal limit points within the geometrically finite…

In this paper we study counting functions representing the number of solutions of systems of linear inequalities which arise in the theory of Diophantine approximation. We develop a method that allows us to explain the random-like behavior…

We prove uniqueness in law for possibly degenerate SDEs having a linear part in the drift term. Diffusion coefficients corresponding to non-degenerate directions of the noise are assumed to be continuous. When the diffusion part is constant…

We address the question of the infinitude of twin and cousin prime pairs from a probabilistic perspective. Our approach partitions the set of integer numbers greater than $2$ in finite intervals of the form $[p_{n-1}^2,p_n^2)$, $p_{n-1}$…

We study the problem of Diophantine approximation on lines in $\mathbb{C}^2$ with numerators and denominators restricted to Gaussian primes. To this end, we develop analogs of well-known results on small fractional parts of $p\gamma$, $p$…

Let $\mathbb{Z}^{ab}$ be the ring of integers of $\mathbb{Q}^{ab}$, the maximal abelian extension of $\mathbb{Q}$. We show that there exists an algorithm to decide whether a system of equations and inequations, with integer coefficients,…

Locality is a property of logics, based on Hanf's and Gaifman's theorems, and that was shown to be very useful in the context of finite model theory. In this paper I present a homotopic variation for locality, namely a Quillen model…

Let k\geq 2 and consider the Diophantine inequality |x_1^k-\alp_2 x_2^k-\alp_3 x_3^k| <\tet. Our goal is to find non-trivial solutions in the variables x_i, 1\leq i\leq 3, all of size about P, assuming that \tet is sufficiently large. We…

We consider the problem of approaching real numbers with rational numbers with prime denominator and with a single numerator allowed for each denominator. We obtain basic results, both probabilistic and deterministic, draw connections to…

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…

We derive new cases of conjectures of Rubin and of Burns--Kurihara--Sano concerning derivatives of Dirichlet $L$-series at $s = 0$ in $p$-elementary extensions of number fields for arbitrary prime numbers $p$. In naturally arising examples…

We investigate the distribution of $p\theta$ modulo 1, where $\theta$ is a complex number which is not contained in $\mathbb{Q}(i)$, and $p$ runs over the Gaussian primes.

A simple corollary of the localization theorem (due to the author and, independently, to Lian-Liu-Yau) is applied to several problems in enumerative geometry. New formulas for Schubert calculus on flag manifolds, due to Kong, and a new…

Diophantine approximation is the problem of approximating a real number by rational numbers. We propose a version of this in which the numerators are approximately related to the denominators by a Laurent polynomial. Our definition is…

We present the first quantum system where Anderson localization is completely described within periodic-orbit theory. The model is a quantum graph analogous to an a-periodic Kronig-Penney model in one dimension. The exact expression for the…

We shall show that, for any given primes $\ell\geq 17$ and $p, q\equiv 1\pmod{\ell}$, the diophantine equation $(x^\ell-1)/(x-1)=p^m q$ has at most four positive integral solutions $(x, m)$ and give its application to odd perfect number…

In this paper, we give a proof of the classical Kronecker limit formulas using the distribution relation of the Eisenstein-Kronecker series. Using a similar idea, we then prove $p$-adic analogues of the Kronecker limit formulas for the…

Using the Thue-Siegel method, we obtain effective improvements on Liouville's irrationality measure for certain one-parameter families of algebraic numbers, defined by equations of the type $(t-a)Q(t)+P(t)=0$. We apply these to some…

We study some problems in metric Diophantine approximation over local fields of positive characteristic.

Let k => 1, m => 1 be small fixed integers, gcd(k, m) = 1. This note develops some techniques for proving the existence of infinitely many primes solutions x = p, and y = q of the linear Diophantine equation y = mx + k.