Related papers: Approximating elements of the middle third Cantor …

Let $C_a$ be the central Cantor set obtained by removing a central interval of length $1-2a$ from the unit interval, and continuing this process inductively on each of the remaining two intervals. We prove that if $\log b/\log a$ is…

In this paper, we show that if we have a sequence of Hadamard triples $\{(N_n,B_n,L_n)\}$ with $B_n\subset \{0,1,..,N_n-1\}$ for $n=1,2,...$, except an extreme case, then the associated Cantor-Moran measure $$ \begin{aligned} \mu =…

The Jarn\'ik-Besicovitch theorem is a fundamental result in metric number theory which concerns the Hausdorff dimension for certain limsup sets. We discuss the analogous problem for liminf sets. Consider an infinite sequence of positive…

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…

In this paper, we obtain the upper bounds for the Hausdorff measures of nodal sets of eigenfunctions with the Robin boundary conditions, i.e., \begin{equation*} {\left\{\begin{array}{l} \triangle u+\lambda u=0,\quad in\quad \Omega,\\…

Let $\cal C$ be a non--degenerate planar curve and for a real, positive decreasing function $\psi$ let $\cal C(\psi)$ denote the set of simultaneously $\psi$--approximable points lying on $\cal C$. We show that $\cal C$ is of Khintchine…

For any positive integer $p\geq 3$, let $A$ be a proper subset of $\{0,1,\ldots, p-1\}$ with $\sharp A=s\geq 2$. Suppose $h: \{0,1,\ldots,s-1\}\to A$ is a one-to-one map which is strictly increasing with $A=\{h(0),h(1),\ldots,h(s-1)\}$. We…

The questions of the measure and finding open intervals in certain sets of sums and products of elements of the middle third Cantor set (or a variant of it), have generated considerable interest recently. A broad general framework that…

Let $F \subseteq [0,1]$ be a set that supports a probability measure $\mu$ with the property that $ |\widehat{\mu}(t)| \ll (\log |t|)^{-A}$ for some constant $ A > 0 $. Let $\mathcal{A}= (q_n)_{n\in \mathbb{N}} $ be a sequence of natural…

We compute the Hausdorff dimension of sets of very well approximable vectors on rational quadrics. We use ubiquitous systems and the geometry of locally symmetric spaces. As a byproduct we obtain the Hausdorff dimension of the set of rays…

The use of Hausdorff measures and dimension in the theory of Diophantine approximation dates back to the 1920s with the theorems of Jarnik and Besicovitch regarding well-approximable and badly-approximable points. In this paper we consider…

Let $G$ be a $p$-adic analytic pro-$p$ group of dimension $d$. We produce an approximate series which descends regularly in strata and whose terms deviate from the lower $p$-series in a uniformly bounded way. This brings to light a new set…

In this paper we initiate a new approach to studying approximations by rational points to points on smooth submanifolds of $\mathbb{R}^n$. Our main result is a convergence Khintchine type theorem for arbitrary nondegenerate submanifolds of…

Let $f:M\to M$ be a homeomorphism over a compact Riemannian manifold, ergodic with respect to a measure $\mu$ defined on the completion of the Borel $\sigma$-algebra and $\mathcal F$ a $f$-invariant one dimensional continuous foliation of…

Let A be a subset of the primes. Let \delta_P(N) = \frac{|\{n\in A: n\leq N\}|}{|\{\text{$n$ prime}: n\leq N\}|}. We prove that, if \delta_P(N)\geq C \frac{\log \log \log N}{(\log \log N)^{1/3}} for N\geq N_0, where C and N_0 are absolute…

Let $W(\p)$ denote the set of $\p$-well approximable points in $\R^d$ and let $K$ be a compact subset of $\R^d$ which supports a measure $\mu$. In this short note, we show that if $\mu$ is an `absolutely friendly' measure and a certain…

We establish effective convergence rates in the Doeblin-Lenstra law, describing the limiting distribution of approximation coefficients arising from continued fraction convergents of a typical real number. More generally, we prove…

We consider (not self-similar) Cantor sets defined by a sequence of piecewise linear functions. We prove that the dimension of the harmonic measure on such a set is strictly smaller than its Hausdorff dimension. Some Hausdorff measure…

Let $\cS_n(\psi_1,...,\psi_n)$ denote the set of simultaneously $(\psi_1,...,\psi_n)$--approximable points in $\R^n$ and $\cSM_n(\psi)$ denote the set of multiplicatively $\psi$--approximable points in $\R^n$. Let $\cM$ be a manifold in…

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.