Related papers: Counting real algebraic numbers with bounded deriv…

We investigate approximation to a given real number by algebraic numbers and algebraic integers of prescribed degree. We deal with both best and uniform approximation, and highlight the similarities and differences compared with the…

The Hausdorff dimension of the set of simultaneously tau well approximable points lying on a curve defined by a polynomial P(X)+alpha, where P(X) is a polynomial with integer coefficients and alpha is in R, is studied when tau is larger…

We count algebraic points of bounded height and degree on the graphs of certain functions analytic on the unit disk, obtaining a bound which is polynomial in the degree and in the logarithm of the multiplicative height. We combine this work…

The distribution of $\alpha p$ modulo one, where $p$ runs over the rational primes and $\alpha$ is a fixed irrational real, has received a lot of attention. It is natural to ask for which exponents $\nu>0$ one can establish the infinitude…

We compute the Hausdorff dimension of the set of simultaneously $q^{-\lambda}$-well approximable points on the Veronese curve in $\mathbb{R}^n$ for $\lambda$ between $\frac{1}{n}$ and $\frac{2}{2n-1}$. For $n=3$, the same result is given…

Building on work of Davenport and Schmidt, we mainly prove two results. The first one is a version of Gel'fond's transcendence criterion which provides a sufficient condition for a complex or $p$-adic number $\xi$ to be algebraic in terms…

Given an increasing integer sequence $(a_n)$, a real number $\alpha$, and a sequence $\psi(n)$, we study the set $W$ of real numbers $\gamma$ for which $a_n\alpha - \gamma$ is a distance less than $\psi(n)$ away from an integer. This is…

We study two polynomial counting questions in arithmetic statistics via a combination of Fourier analytic and arithmetic methods. First, we obtain new quantitative forms of Hilbert's Irreducibility Theorem for degree $n$ polynomials $f$…

We are studying the Diophantine exponent \mu_{n,l}$ defined for integers 1 \leq l < n and a vector \alpha \in \mathbb{R}^n by letting \mu_{n,l} = \sup{\mu \geq 0: 0 < ||x \cdot \alpha|| < H(x)^{-\mu} for infinitely many x \in C_{n,l} \cap…

Let $n\in\mathbb{N}$ be fixed, $Q>1$ be a real parameter and $\mathcal{P}_n(Q)$ denote the set of polynomials over $\mathbb{Z}$ of degree $n$ and height at most $Q$. In this paper we investigate the following counting problems regarding…

Let Q be an infinite set of positive integers. Denote by W_{\tau, n}(Q) (resp. W_{\tau, n}) the set of points in dimension n simultaneously \tau--approximable by infinitely many rationals with denominators in Q (resp. in N*). A non--trivial…

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}},\,…

The height of an algebraic number $\alpha$ is a measure of how arithmetically complicated $\alpha$ is. We say $\alpha$ is totally $p$-adic if the minimal polynomial of $\alpha$ splits completely over the field $\mathbb{Q}_p$ of $p$-adic…

We consider an algebraic variety and its foliation, both defined over a number field. We prove upper bounds for the geometric complexity of the intersection between a leaf of the foliation and a subvariety of complementary dimension (also…

We prove that for any real polynomial $f(x) \in\mathbb{R} [x]$ the set $$ \{\alpha \in \mathbb{R}: \liminf_{n\to \infty} n\log n ||\alpha f(n)|| >0\} $$ has positive Hausdorff dimension. Here $||\xi ||$ means the distance from $\xi $ to the…

Point counting estimates are a key stepping stone to various results in metric Diophantine approximation. In this paper we use the quantitative non-divergence estimates originally developed by Kleinbock and Margulis to improve lower bounds…

For $\theta$ a non-algebraic point on a quasi projective variety over a number field, I prove that $\theta$ has an approximation by a series of algebraic points of bounded height and degree which is essentially best possible. Applications…

Cantor gave in his fundamental article an elegant proof of the countability of real algebraic numbers based on a positive integer height, denoted by him as N, of integer and irreducible polynomials of given degree (denoted by him as n) with…

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…

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…