Related papers: $HD(M\setminus L)>0.353$
We construct a family of non-PCF dendrites $K$ in a plane, such that in each of them all subarcs have the same Hausdorff dimension $s$, while the set of $s$-dimensional Hausdorff measures of subarcs connecting the given point and a…
We study maximal averages associated with singular measures on $\rr$. Our main result is a construction of singular Cantor-type measures supported on sets of Hausdorff dimension $1 - \epsilon$, $0 \leq \epsilon < {1/3}$ for which the…
In a general measure space $(X,\mathcal L,\lambda)$, a characterization of weakly null sequences in $L_\infty (X,\mathcal L,\lambda)$ ($u_k \rightharpoonup 0$) in terms of their pointwise behaviour almost everywhere is derived from the…
We prove that a homeomorphism f of a compact metric space X satisfies the L-shadowing property if and only if its induced hyperspace homeomorphism also satisfies the L-shadowing property. In the proof, assuming only the L-shadowing…
In this paper we prove some lower bounds on the Hausdorff dimension of sets of Furstenberg type. Moreover, we extend these results to sets of generalized Furstenberg type, associated to doubling dimension functions. With some additional…
We use the galaxy angular power spectrum at $z\sim0.5-1.2$ from the Canada-France-Hawaii-Telescope Legacy Survey Wide fields (CFHTLS-Wide) to constrain separately the total neutrino mass $\sum{m_\nu}$ and the effective number of neutrino…
In this paper we introduce and study a certain intricate Cantor-like set $C$ contained in unit interval. Our main result is to show that the set $C$ itself, as well as the set of dissipative points within $C$, both have Hausdorff dimension…
In 1969 H\"ochster proved that for every quasi-compact T1-space $X$ we can find a commutative ring $R$ such that $X$ is homeomorphic to the maximal spectrum $\mathrm{Specm}(R)$ of $R$. This result implies the existence of a commutative ring…
An important theorem of geometric measure theory (first proved by Besicovitch and Davies for Euclidean space) says that every analytic set of non-zero $s$-dimensional Hausdorff measure $\mathcal H^s$ contains a closed subset of non-zero…
Let E be a metric space. We introduce a notion of connectedness index of E, which is the Hausdor? dimension of the union of non-trivial connected components of E. We show that the connectedness index of a fractal cube E is strictly less…
The experiments at the Large Hadron Collider (LHC) have pushed the limits on masses of supersymmetric particles beyond the $\sim$TeV scale. This compromises naturalness of the simplest supersymmetric extension of the Standard Model, the…
In this paper, we study inhomogeneous Diophantine approximation over the completion $K_v$ of a global function field $K$ (over a finite field) for a discrete valuation $v$, with affine algebra $R_v$. We obtain an effective upper bound for…
The concept of sliding Shilnikov connection has been recently introduced and represents an important notion in Filippov systems, because its existence implies chaotic behavior on an invariant subset of the system. The investigation of its…
In this paper, we will introduce and study the lower moving digit mean $\underline{M}(x)$ and the upper moving digit mean $\overline{M}(x)$ of $x\in[0,1]$ in $p$-adic expansion, where $p\geq2$ is an integer. Moreover, the Hausdorff…
The aim of this work is to give a pointfree description of the Cantor set. It can be shown that the Cantor set is homeomorphic to the $p$-adic integers $\mathbb{Z}_{p}:=\{x\in\mathbb{Q}_{p}: |x|_p\leq 1\}$ for every prime number $p$. To…
Let $\varphi_0$ be a $C^2$-conservative diffeomorphism of a compact surface $S$ and let $\Lambda_0$ be a mixing horseshoe of $\varphi_0$. Given a smooth real function $f$ defined in $S$ and some diffeomorphism $\varphi$, close to…
We establish sharp bounds for the Hausdorff dimension of sets of irrational numbers in $(0,1)$ whose digits in the $N$-expansion are either uniformly bounded or tend to infinity. For sets with digits bounded by an integer $M \ge N$, we…
We prove a structure theorem for any $n$-rectifiable set $E\subset \mathbb{R}^{n+1}$, $n\ge 1$, satisfying a weak version of the lower ADR condition, and having locally finite $H^n$ ($n$-dimensional Hausdorff) measure. Namely, that…
In this paper, we find a full Lebesgue measure set of frequencies $\check \II\subset [0,1]\setminus \Q$ such that for any $(\alpha,\lambda)\in \check \II\times [24,\infty)$, the Hausdorff and box dimensions of the spectrum of the Sturmian…
The strongly coupled heterotic M-theory vacuum for both the observable and hidden sectors of the $B-L$ MSSM theory is reviewed, including a discussion of the "bundle" constraints that both the observable sector $SU(4)$ vector bundle and the…