We study the distribution of harmonic measure on connected Julia sets of unicritical polynomials. Harmonic measure on a full compact set in ℂ is always concentrated on a set which is porous for a positive density of scales. We prove that there is a topologically generic set $\mathcal{A}$ in the boundary of the Mandelbrot set such that for every $c\in \mathcal{A}$ ,$β$>0, and$λ$∈(0,1), the corresponding Julia set is a full compact set with harmonic measure concentrated on a set which is not$β$-porous in scale$λ$^{$n$}for$n$from a set with positive density amongst natural numbers.

The present paper considers volume formulæ, as well as trigonometric identities, that hold for a tetrahedron in 3-dimensional spherical space of constant sectional curvature +1. The tetrahedron possesses a certain symmetry: namely rotation of angle$π$in the middle points of a certain pair of its skew edges.

We study correlation bounds under pairwise independent distributions for functions with no large Fourier coefficients. Functions in which all Fourier coefficients are bounded by$δ$are called$δ$-$uniform$. The search for such bounds is motivated by their potential applicability to hardness of approximation, derandomization, and additive combinatorics.

On montre que la fonction maximale centrée de Hardy–Littlewood,$M$, sur les espaces hyperboliques réels $\mathbb{H}^{n} = \mathbb{R}^{{\mathchoice {\raise .17ex\hbox {$\scriptstyle +$}}{\raise .17ex\hbox {$\scriptstyle +$}}{\raise .1ex\hbox {$\scriptscriptstyle +$}}{\scriptscriptstyle +}}} \times \mathbb{R}^{n - 1}$ , satisfait l’inégalité de type faible $\| M f \|_{L^{1, \infty}} \leq A (n \log {n}) \| f\|_{1}$ pour toute$f$∈$L$^{1}(ℍ^{$n$}), où$A$>0 est une constante indépendante de la dimension$n$.

Let Ω be a Greenian domain in ℝ^{$d$},$d$≥2, or—more generally—let Ω be a connected $\mathcal{P}$ -Brelot space satisfying axiom D, and let$u$be a numerical function on Ω, $u\not\equiv\infty$ , which is locally bounded from below. A short proof yields the following result: The function$u$is the infimum of its superharmonic majorants if and only if each set {$x$:$u$($x$)>$t$},$t$∈ℝ, differs from an analytic set only by a polar set and $\int u\,d\mu_{x}^{V}\le u(x)$ , whenever$V$is a relatively compact open set in Ω and$x$∈$V$.