Related papers: Badly approximable points for diagonal approximati…

In this paper, we consider a problem of counting rational points near self-similar sets. Let $n\geq 1$ be an integer. We shall show that for some self-similar measures on $\mathbb{R}^n$, the set of rational points $\mathbb{Q}^n$ is…

Let $X = G/\Gamma$, where $G$ is a Lie group and $\Gamma$ is a lattice in $G$, and let $U$ be a subset of $X$ whose complement is compact. We use the exponential mixing results for diagonalizable flows on $X$ to give upper estimates for the…

The purpose of this article is twofold. On the one hand, we prove asymptotic formulas for the quantitative distribution of rational points on any smooth non-split projective quadratic surface. We obtain the optimal error term for the real…

We study the Hausdorff measure and dimension of the set of intrinsically simultaneously $\psi$-approximable points on a curve, surface, etc., given as a graph of integer valued polynomials. We obtain complete answers to these questions for…

Infinite-dimensional, holomorphic functions have been studied in detail over the last several decades, due to their relevance to parametric differential equations and computational uncertainty quantification. The approximation of such…

A badly approximable system of affine forms is determined by a matrix and a vector. We show Kleinbock's conjecture for badly approximable systems of affine forms: for any fixed vector, the set of badly approximable systems of affine forms…

We reconsider the Euler-Lagrange equation for the Skyrme model in the hedgehog ansatz and study the analytical properties of the solitonic solution. In view of the lack of a closed form solution to the problem, we work on approximate…

We consider discrete best approximation problems in the setting of tropical algebra, which is concerned with the theory and application of algebraic systems with idempotent operations. Given a set of input--output pairs of an unknown…

Given a weight vector $\tau=(\tau_{1}, \dots, \tau_{n}) \in \mathbb{R}^{n}_{+}$ with each $\tau_{i}$ bounded by certain constraints, we obtain a lower bound for the Hausdorff dimension of the set of $\tau$-approximable points points over a…

We consider intersections of n diagonal forms of degrees k 1 < $\bullet$ $\bullet$ $\bullet$ < kn, and we prove an asymptotic formula for the number of rational points of bounded height on these varieties. The proof uses the…

In 1984, Kurt Mahler posed the following fundamental question: How well can irrationals in the Cantor set be approximated by rationals in the Cantor set? Towards development of such a theory, we prove a Dirichlet-type theorem for this…

We consider approximation of diameter of a set $S$ of $n$ points in dimension $m$. E$\tilde{g}$ecio$\tilde{g}$lu and Kalantari \cite{kal} have shown that given any $p \in S$, by computing its farthest in $S$, say $q$, and in turn the…

Fix an irrational number $\theta$. For a real number $\tau >0$, consider the numbers $y$ satisfying that for all large number $Q$, there exists an integer $1\leq n\leq Q$, such that $\|n\theta-y\|<Q^{-\tau}$, where $\|\cdot\|$ is the…

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…

The set B of geodesic rays avoiding a suitable obstacle in a complete negatively curved Riemannian manifold determines a spectrum S. While various properties of this spectrum are known, we define and study dimension functions on S in terms…

A classical result of Kaufman states that, for each $\tau>1,$ the set of well approximable numbers \[ E(\tau)=\{x\in\mathbb{R}: \|qx\| < |q|^{-\tau} \text{ for infinitely many integers q}\} \] is a Salem set with Hausdorff dimension…

We calculate the measure and Hausdorff dimension of sets of matrices over fields of formal power series with good approximation properties for a restricted set of denominators.

In the optimization of convex domains under a PDE constraint numerical difficulties arise in the approximation of convex domains in $\mathbb{R}^3$. Previous research used a restriction to rotationally symmetric domains to reduce shape…

We study the complexity of approximating complex zero sets of certain $n$-variate exponential sums. We show that the real part, $R$, of such a zero set can be approximated by the $(n-1)$-dimensional skeleton, $T$, of a polyhedral…

Let C be two times continuously differentiable curve in R^2 with at least one point at which the curvature is non-zero. For any i,j > 0 with i+j =1, let Bad(i,j) denote the set of points (x,y) in R^2 for which max {||qx ||^{1/i},…