Related papers: Diophantine approximation on curves
Metric Diophantine approximation in its classical form is the study of how well almost all real numbers can be approximated by rationals. There is a long history of results which give partial answers to this problem, but there are still…
In quantum theory on curved backgrounds, Heisenberg's uncertainty principle is usually discussed in terms of ensemble variances and flat-space commutators. Here we take a different, preparation-based viewpoint tailored to sharp position…
Recently, Adiceam, Beresnevich, Levesley, Velani and Zorin proved a quantitative version of the convergence case of the Khintchine-Groshev theorem for nondegenerate manifolds, motivated by applications to interference alignment. In the…
In this paper, we investigate the Hausdorff dimension of naturally occurring sets of inhomogeneous well-approximable points with a sequence of real invertible matrices $\mathcal{A}=(A_n)_{n\in\mathbb{N}}$. Specifically, for a given point…
This work is motivated by problems on simultaneous Diophantine approximation on manifolds, namely, establishing Khintchine and Jarnik type theorems for submanifolds of R^n. These problems have attracted a lot of interest since Kleinbock and…
The theory of uniform Diophantine approximation concerns the study of Dirichlet improvable numbers and the metrical aspect of this theory leads to the study of the product of consecutive partial quotients in continued fractions. It is known…
We show that the Continuum Hypothesis implies that for every $0<d_1\leq d_2<n$ the measure spaces $(\RR^n,\iM_{\iH^{d_1}},\iH^{d_1})$ and $(\RR^n,\iM_{\iH^{d_2}},\iH^{d_2})$ are isomorphic, where $\iH^d$ is $d$-dimensional Hausdorff measure…
Considerable attention has been given to the study of the arithmetic sum of two planar sets. We focus on understanding the measure and dimension of $A+\Gamma:=\left\{a+v:a\in A, v\in \Gamma \right\}$ when $A\subset \mathbb{R}^2$ and…
The main objective of this paper is to prove a Khintchine type theorem for divergence for linear Diophantine approximation on non-degenerate manifolds, which completes earlier results for convergence.
We introduce a new notion of a harmonic measure for a $d$-dimensional set in $\R^n$ with $d<n-1$, that is, when the codimension is strictly bigger than 1. Our measure is associated to a degenerate elliptic PDE, it gives rise to a…
In 1995, Hill and Velani introduced the shrinking targets theory. Given a dynamical system $([0,1],T)$, they investigated the Hausdorff dimension of sets of points whose orbits are close to some fixed point. In this paper, we study the sets…
In a landmark paper, D.Y. Kleinbock and G.A. Margulis established the fundamental Baker-Sprindzuk conjecture on homogeneous Diophantine approximation on manifolds. Subsequently, there has been dramatic progress in this area of research.…
Due to its many applications, \emph{curve simplification} is a long-studied problem in computational geometry and adjacent disciplines, such as graphics, geographical information science, etc. Given a polygonal curve $P$ with $n$ vertices,…
Let $(X, d)$ be a compact metric space, and let $Q \subset X$ be countable. Given functions $R: Q \to \mathbb{R}^+$ and $\phi: \mathbb{R}^+ \to \mathbb{R}^+$, we consider the set $E(Q, R, \phi)$ of points $x \in X$ that ``hit'' the…
In this survey article, we explore a central theme in Diophantine approximation inspired by a celebrated result of Besicovitch on the Hausdorff dimension of well approximable real numbers. We outline some of the key developments stemming…
Given a compact K\"ahler manifold $(X,\omega)$, due to the work of Darvas-Di Nezza-Lu, the space of singularity types of $\omega$-psh functions admits a natural pseudo-metric $d_\mathcal S$ that is complete in the presence of positive mass.…
Gallagher's theorem is a sharpening and extension of the Littlewood conjecture that holds for almost all tuples of real numbers. We provide a fibre refinement, solving a problem posed by Beresnevich, Haynes and Velani in 2015. Hitherto,…
We fill a gap in the study of the Hausdorff dimension of the set of exact approximation order considered by Fregoli [Proc. Amer. Math. Soc. 152 (2024), no. 8, 3177--3182].
The convex and metric structures underlying probabilistic physical theories are generally described in terms of base normed vector spaces. According to a recent proposal, the purely geometrical features of these spaces are appropriately…
A Hausdorff measure version of W.M. Schmidt's inhomogeneous, linear forms theorem in metric number theory is established. The key ingredient is a `slicing' technique motivated by a standard result in geometric measure theory. In short,…