Related papers: Fractal percolation is unrectifiable

We consider a fractal refinement of Carleson's problem for pointwise convergence of solutions to the periodic Schr\"odinger equation to their initial datum. For $\alpha \in (0,d]$ and \[ s < \frac{d}{2(d+1)} (d + 1 - \alpha), \] we find a…

Let A be an idempotent algebra on a 3-element domain D that omits a G-set for a factor. Suppose A is not \alpha\beta-projective (for some alpha, beta subsets of D) and is not collapsible. It follows that A is switchable. We prove that, for…

In this paper we consider the problem $-\Delta u=|x|^{\alpha} F(u)$ in $R^N$, with $\alpha>0$ and $N\ge3$. Under some assumptions on $F$ we deduce the existence of nonradial solutions which bifurcate from the radial one when $\alpha$ is an…

In the paper, we investigate the fine multifractal spectrum of a class of self-affine Moran sets with fixed frequencies, and we prove that under certain separation conditions, the fine multifractal spectrum $H(\alpha)$ is given by the…

We prove that Bernoulli convolutions are absolutely continuous provided the parameter lambda is an algebraic number sufficiently close to 1 depending on the Mahler measure of lambda.

In this paper, we investigate the fractal uncertainty principle (FUP) for discrete Cantor sets, which are determined by an alphabet from a base of digits. Consider the base of M digits and the alphabets of cardinality A such that all the…

A partition of the positive integers into sets $A$ and $B$ {\em avoids} a set $S\subset\N$ if no two distinct elements in the same part have a sum in $S$. If the partition is unique, $S$ is {\em uniquely avoidable.} For any irrational…

This note treats several problems for the fractional perimeter or $s$-perimeter on the sphere. The spherical fractional isoperimetric inequality is established. It turns out that the equality cases are exactly the spherical caps.…

We consider integer sequences that satisfy a recursion of the form $x_{n+1} = P(x_n)$ for some polynomial $P$ of degree $d > 1$. If such a sequence tends to infinity, then it satisfies an asymptotic formula of the form $x_n \sim A…

We examine the stability of a 1D electrical transmission line in the simultaneous presence of PT-symmetry and fractionality. The array contains a binary gain/loss distribution $\gamma_{n}$ and a fractional Laplacian characterized by a…

We prove a fractal uncertainty principle with exponent $\frac{d}{2} - \delta + \varepsilon$, $\varepsilon > 0$, for Ahlfors--David regular subsets of $\mathbb R^d$ with dimension $\delta$ which satisfy a suitable "nonorthogonality…

In previous papers by A. Kameyama and by J. Kigami distances on fractals have been discussed having two different but similar properties. One property is that the maps defining the fractal are Lipschitz of prescribed constants less than 1,…

We extend Smirnov's proof of the existence and conformal invariance of the scaling limit of critical site-percolation on the triangular lattice to particular sequences of periodic graphs with more arbitrary large-scale structure, obtained…

We prove that for Bernoulli percolation on a graph $\mathbb{Z}^2\times\{0,\dots,k\}$ ($k\ge 0$), there is no infinite cluster at criticality, almost surely. The proof extends to finite range Bernoulli percolation models on $\mathbb{Z}^2$…

We evaluate the percolation threshold values for a realistic model of continuum segregated systems, where random spherical inclusions forbid the percolating objects, modellized by hard-core spherical particles surrounded by penetrable…

We build a multifractal object and use it as a support to study percolation. We identify some differences between percolation in a multifractal and in a regular lattice. We use many samples of finite size lattices and draw the histogram of…

The fact that galaxy distribution exhibits fractal properties is well established since twenty years. Nowadays, the controversy concerns the range of the fractal regime, the value of the fractal dimension and the eventual presence of a…

Fractional derivatives are generalization to classical integer-order derivatives. The rules which are true for classical derivative need not hold for the fractional derivatives, for example, we cannot simply add the fractional orders…

This article examines the Thomae function, a paradigmatic example of a function that is continuous on the irrationals and discontinuous elsewhere. Defined for a parameter $\theta>0$, it exhibits a rich self-similar structure and intriguing…

We address the question of whether a reflecting stationary set may be partitioned into two or more reflecting stationary subsets, providing various affirmative answers in ZFC. As an application to singular cardinals combinatorics, we infer…