Related papers: Weighted twisted inhomogeneous Diophantine approxi…

We prove that uniformly disconnected subsets of metric measure spaces with controlled geometry (complete, Ahlfors regular, supporting a Poincare inequality, and a mild topological condition) are contained in a quasisymmetric arc. This…

Let $d_1$, $d_2$, ... be a sequence of positive numbers that converges to zero. A generalization of Steinhaus' theorem due to Weil implies that, if a subset of a homogeneous Riemannian manifold has no pair of points at distances $d_1$,…

The Doob convergence theorem implies that the set of divergence of any martingale has measure zero. We prove that, conversely, any $G\_{\delta\sigma}$ subset of the Cantor space with Lebesgue-measure zero can be represented as the set of…

The classical Khintchine and Jarn\'ik theorems, generalizations of a consequence of Dirichlet's theorem, are fundamental results in the theory of Diophantine approximation. These theorems are concerned with the size of the set of real…

We prove the hyperplane absolute winning property of weighted inhomogeneous badly approximable vectors in $\mathbb{R}^d$. This answers a question by Beresnevich--Nesharim--Yang and extends the main result of [Geometric and Functional…

Let $0<p\leq 1$, and let $\omega:\mathbb N^2 \to [1,\infty)$ be an almost monotone weight. Let $\mathbb H$ be the closed right half plane in the complex plane. Let $\widetilde a$ be a complex valued function on $\mathbb H^2$ such that…

The inhomogeneous Groshev type theory for dual Diophantine approximation on manifolds is developed. In particular, the notion of nice manifolds is introduced and the divergence part of the theory is established for all such manifolds. Our…

By employing an accelerated weighting method, we establish arbitrary polynomial and exponential pointwise convergence for multiple ergodic averages under general balancing conditions in both discrete and continuous settings, including…

We give an elementary proof of a recent metrical Diophantine result by D. Kleinbock related to badly approximable vectors in affine subspaces.

Let $\Gamma = Z A +Z^n$ be a dense subgroup with rank $n+1$ in $R^n$ and let $\omega(A)$ denote the exponent of uniform simultaneous rational approximation to the point $A$. We show that for any real number $v\ge \omega(A)$, the Hausdorff…

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 L\"uroth expansion of a real number $x\in (0,1]$ is the series \[ x= \frac{1}{d_1} + \frac{1}{d_1(d_1-1)d_2} + \frac{1}{d_1(d_1-1)d_2(d_2-1)d_3} + \cdots, \] with $d_j\in\mathbb{N}_{\geq 2}$ for all $j\in\mathbb{N}$. Given $m\in…

We extend almost everywhere convergence in Wiener-Wintner ergodic theorem for $\sigma$-finite measure to a generally stronger almost uniform convergence and present a larger, universal, space for which this convergence holds. We then extend…

Let (X,d) be a metric space and (\Omega, d) a compact subspace of X which supports a non-atomic finite measure m. We consider `natural' classes of badly approximable subsets of \Omega. Loosely speaking, these consist of points in \Omega…

The almost sure convergence of ergodic averages in Birkhoff's pointwise ergodic theorem is known to fail in the finitely additive setting. We introduce a natural reformulation of almost sure convergence suitable for finitely additive…

We examine Fourier frames and, more generally, frame measures for different probability measures. We prove that if a measure has an associated frame measure, then it must have a certain uniformity in the sense that the weight is distributed…

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 establish effective versions of Oppenheim's conjecture for generic inhomogeneous quadratic forms. We prove such results for fixed shift vectors and generic quadratic forms. When the shift is rational we prove a counting result which…

The Riemann Mapping Theorem states existence of a conformal homeomorphism $\varphi$ of a simply connected plane domain $\Omega\subset\mathbb C$ with non-empty boundary onto the unit disc $\mathbb D\subset \mathbb C$. In the first part of…

The Bernstein approximation problem is to determine whether or not the space of all polynomials is dense in a given weighted $C_0$-space on the real line. A theorem of L. de Branges characterizes non--density by existence of an entire…