Related papers: Diophantine Approximations on Definable Sets

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…

We provide polynomial upper bounds for the minimal sizes of distal cell decompositions in several kinds of distal structures, particularly weakly $o$-minimal and $P$-minimal structures. The bound in general weakly $o$-minimal structures…

In the field of algebraic systems biology, the number of minimal polynomial models constructed using discretized data from an underlying system is related to the number of distinct reduced Gr\"obner bases for the ideal of the data points.…

We give elementary proof of stronger versions of several recent results on intrinsic Diophantine approximation on rational quadric hypersurfaces $X\subset \mathbb{P}^n(\mathbb{R})$. The main tool is a refinement of the simplex lemma, which…

We study the set of tangent limits at a given point to a set definable in any o-minimal structure by characterizing the set of exceptional rays in the tangent cone to the set at that point and investigating the set of tangent limits along…

Inspired by a problem proposed by Mahler, we will address the following related question, 'How well can irrationals in a missing digit set be approximated by rationals with polynomial denominators?' and prove some related results. To…

We investigate the problem of best simultaneous Diophantine approximation under a constraint on the denominator, as proposed by Jurkat. New lower estimates for optimal approximation constants are given in terms of critical determinants of…

We characterize the notion of definable compactness for topological spaces definable in o-minimal structures, answering questions of Peterzil and Steinhorn (1999) and Johnson (2018). Specifically, we prove the equivalence of various…

We develop tame topology over dp-minimal structures equipped with definable uniformities satisfying certain assumptions. Our assumptions are enough to ensure that definable sets are tame: there is a good notion of dimension on definable…

We establish an explicit asymptotic formula for the number of rational solutions of intrinsic Diophantine inequalities on simply-connected simple algebraic groups, at arbitrarily small scales.

In this paper we study a quantitative notion of exactness within Diophantine approximation. Given $\Psi:(0,\infty)\to (0,\infty)$ and $\omega:(0,\infty)\to (0,1)$ satisfying $\lim_{q\to\infty}\omega(q)=0$, we study the set of points, which…

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 quantify the density of rational points in the unit sphere $S^n$, proving analogues of the classical theorems on the embedding of $\q^n$ into $\r^n$. Specifically, we prove a Dirichlet theorem stating that every point $\alpha \in S^n$ is…

We construct a class of multiple Legendre polynomials and prove that they satisfy an Ap\'ery-like recurrence. We give new upper bounds of the approximation measures of logarithms of rational numbers by algebraic numbers of bounded degree.…

We generalize Dirichlet's diophantine approximation theorem to approximating any real number $\alpha$ by a sum of two rational numbers $\frac{a_1}{q_1} + \frac{a_2}{q_2}$ with denominators $1 \leq q_1, q_2 \leq N$. This turns out to be…

In this paper we prove that for any definable subset $X\subset \mathbb{R}^{n}$ in a polynomially bounded o-minimal structure, with $dim(X)<n$, there is a finite set of regular projections (in the sense of Mostowski ). We give also a weak…

Except for a limited number of cases, a complete classification of the Diophantine sets of polynomial rings and fields of rational functions seems out of reach at present. We contribute to this problem by proving that several natural sets…

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…

The paper deals with Henselian valued field with analytic structure. Actually, we are focused on separated analytic structures, but the results remain valid for strictly convergent analytic ones as well. A classical example of the latter is…

We outline a proof of an analogue of Khintchine's Theorem in R^2, where the ordinary height is replaced by a distance function satisfying an irrationality condition as well as certain decay and symmetry conditions.