Related papers: A lower bound on the quantitative version of the t…

The present paper studies a quantitative version of the transversality theorem. More precisely, given a continuous function $g\in \mathcal{C}([0,1]^d,\mathbb{R}^m)$ and a global smooth manifold $W\subset \mathbb{R}^m$ of dimension $p$, we…

In this paper, we establish a refined transversality theorem on linear perturbations from a new perspective of Hausdorff measures. Furthermore, we give its applications not only to singularity theory but also to multiobjective optimization.

We prove a quantitative distortion theorem for iterated function systems that generate sets of continued fractions. As a consequence, we obtain upper and lower bounds on the Hausdorff dimension of any set of real or complex continued…

We give an estimate for the volume of an analytic variety (or more generally the mass of a positive closed current) close to a real submanifold $M$. Applications are given to the Hausdorff measure of the intersection of the variety with $M$…

We prove an appropriate sharp quantitative reverse H\"older inequality for the $C_p$ class of weights from which we obtain as a limiting case the sharp reverse H\"older inequality for the $A_\infty$ class of weights. We use this result to…

Let ${\cal M} \subset \mathbb{R}^d$ be a compact, smooth and boundaryless manifold with dimension $m$ and unit reach. We show how to construct a function $\varphi: \mathbb{R}^d \rightarrow \mathbb{R}^{d-m}$ from a uniform…

Le th\'eor\`eme principal de cet article est un r\'esultat de transversalit\'e quantitative relatif \`a une sous-vari\'et\'e. The main theorem of this paper is a result of estimated transversality with respect to a given submanifold.

We study the class of transversal submanifolds. We characterize their blow-ups at transversal points and prove a negligibility theorem for their "generalized characteristic set", with respect to the Carnot-Carath\'eodory Hausdorff measure.…

We provide a sharp monotonicity theorem about the distribution of subharmonic functions on manifolds, which can be regarded as a new, measure theoretic form of the uncertainty principle. As an illustration of the scope of this result, we…

A fundamental challenge within the metric theory of continued fractions involves quantifying sets of real numbers, when represented using continued fractions, exhibit partial quotients that grow at specific rates. For any positive function…

We study sharpness of various generalizations of Frostman's lemma. These generalizations provide better estimates for the lower Hausdorff dimension of measures. As a corollary, we prove that if a generalized anisotropic gradient…

The paper establishes a sharp quantitative estimate for the $(d-1)$-Hausdorff measure of the critical set of $\mathcal{C}^1$ vector-valued functions on $\mathbb{R}^d$. Additionally, we prove that for a generic $\mathcal{C}^2$ function where…

We consider harmonic functions in the unit ball of $\mathbb{R}^{n+1}$ that are unbounded near the boundary but can be estimated from above by some (rapidly increasing) radial weight $w$. Our main result gives some conditions on $w$ that…

We prove a quantitative version of the Duffin-Schaeffer conjecture with an almost sharp error term. Precisely, let $\psi:\mathbb{N}\to[0,1/2]$ be a function such that the series $\sum_{q=1}^\infty \varphi(q)\psi(q)/q$ diverges. In addition,…

We analyze the detection and classification of singularities of functions $f = \chi_B$, where $B \subset \mathbb{R}^d$ and $d = 2,3$. It will be shown how the set $\partial B$ can be extracted by a continuous shearlet transform associated…

In this paper we study sectional curvature bounds for Riemannian manifolds with density from the perspective of a weighted torsion free connection introduced recently by the last two authors. We develop two new tools for studying weighted…

In this paper we derive estimates for linear potentials that hold away from thin subsets. And, inspired by the celebrated work of Huber, we verify that, for a subset that is thin at a point, there is always a geodesic that reaches to the…

This paper studies estimation of and inference on a distribution function $F$ that is concave on the nonnegative half line and admits a density function $f$ with potentially unbounded support. When $F$ is strictly concave, we show that the…

In this note we define a $C^1$ function $F:[0,M]^2\to [0,2]$ that satisfies that its set of critical values has positive measure. This function provides an example, easier than those that usually appear in the literature, of how the order…

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…