arXiv subjects

Najmeddine Attia

Publications and source records attributed to Najmeddine Attia.

8 recordsLinked to original sources

arXiv2026

Projection Theorems for $Φ$-Intermediate Dimensions

$Φ$-intermediate dimensions interpolate between Hausdorff and box-counting dimensions by restricting admissible coverings to scale windows of the form $[Φ(r),r]$. Using a family of $Φ$-dependent kernels, we develop a potential-theoretic framework that characterizes these dimensions in terms of capacities and leads to associated $Φ$-dimension profiles. This framework provides effective tools for obtaining lower bounds from uniform potential estimates. As an application, we prove Marstrand--Mattila type projection theorems, showing that for $γ_{n,m}$-almost all $m$-dimensional subspaces $V$, the $Φ$-intermediate dimensions of $π_V E$ coincide with deterministic profile values depending only on $E$ and $m$. We also discuss consequences for continuity at the Hausdorff end-point and for the box dimensions of typical projections.

math.MG↗

arXiv2024

On the Hewitt Stromberg dimension of product sets

In this paper, we construct new multifractal measures, on the Euclidean space $\mathbb{R}^n$, in a similar manner to Hewitt-Stomberg measures but using the class of all $n$-dimensional half-open binary cubes of covering sets in the definition rather than the class of all balls. As an application we shall be concerned with evaluation of Hewitt-Stromberg dimension of cartesian product sets by means of the dimensions of their components.

math.CA↗

arXiv2024

A note on the generalized Hausdorff and packing measures of product sets in metric space

Let $μ$ and $ν$ be two Borel probability measures on two separable metric spaces $\X$ and $\Y$ respectively. For $h, g$ be two Hausdorff functions and $q\in \R$, we introduce and investigate the generalized pseudo-packing measure ${\RRR}_μ^{q, h}$ and the weighted generalized packing measure ${\QQQ}_μ^{q, h}$ to give some product inequalities : $${\HHH}_{μ\times ν}^{q, hg}(E\times F) \le {\HHH}_μ^{q, h}(E) \; {\RRR}_ν^{q, g}(F) \le {\RRR}_{μ\times ν}^{q, hg}(E\times F)$$ and $${\PPP}_{μ\times ν}^{q, hg}(E\times F) \le {\QQQ}_μ^{q, h}(E) \; {\PPP}_ν^{q, g}(F) $$ for all $E\subseteq \X$ and $F\subseteq \Y$, where ${\HHH}_μ^{q, h}$ and ${\PPP}_μ^{q, h}$ is the generalized Hausdorff and packing measures respectively. As an application, we prove that under appropriate geometric conditions, there exists a constant $c$ such that $${\HHH}_{μ\times ν}^{q, hg}(E\times F) \le c\, {\HHH}_μ^{q, h}(E) \; {\PPP}_ν^{q, g}(F) $$ $${\HHH}_μ^{q, h}(E) \; {\PPP}_ν^{q, g}(F) \le c\, {\PPP}_μ^{q, hg}(E \times F) $$ $${\PPP}_{μ\times ν}^{q, hg}(E\times F) \le c\, {\PPP}_μ^{q, h}(E) \; {\PPP}_ν^{q, g}(F). $$ These appropriate inequalities are more refined than well know results since we do no assumptions on $μ, ν, h$ and $g$.

math.CA↗

arXiv2024

On a class of Hausdorff measure of cartesian product sets

In this paper, we study, in a separable metric space, a class of Hausdorff measures $\mathcal{H}_μ^{q, ξ}$ defined using a measure $μ$ and a premeasure $ξ$. We discuss a Hausdorff structure of product sets. Weighted Hausdorff measures $\mathcal{W}_μ^{q, ξ}$ appeared as an important tool when studying the product sets. When $μ$ and $ξ$ are blanketed, we prove that $\mathcal{H}_μ^{q, ξ} = \mathcal{W}_μ^{q, ξ}$. As an application, the case when $ξ$ is defined as the Hausdorff function is considered.

math.CA↗

arXiv2022

Multifractal analysis and Erdös-Rényi laws of large numbers for branching random walks in $\R^d$

We revisit the multifractal analysis of $\R^d$-valued branching random walks averages by considering subsets of full Hausdorff dimension of the standard level sets, over each infinite branch of which a quantified version of the Erdös-Rényi law of large numbers holds. Assuming that the exponential moments of the increments of the walks are finite, we can indeed control simultaneously such sets when the levels belong to the interior of the compact convex domain $I$ of possible levels, i.e. when they are associated to so-called Gibbs measures, as well as when they belong to the subset $(\partial{I})_{\mathrm{crit}}$ of $\partial I$ made of levels associated to ``critical'' versions of these Gibbs measures. It turns out that given such a level of one of these two types, the associated Erdös-Rényi LLN depends on the metric with which is endowed the boundary of the underlying Galton-Watson tree. To extend our control to all the boundary points in cases where $\partial I\neq (\partial{I})_{\mathrm{crit}}$, we slightly strengthen our assumption on the distribution of the increments to exhibit a natural decomposition of $\partial I\setminus (\partial{I})_{\mathrm{crit}}$ into at most countably many convex sets $J$ of affine dimension $\le d-1$ over each of which we can essentially reduce the study to that of interior and critical points associated to some $\R^{\dim J}$-valued branching random~walk.

math.PR↗

arXiv2019

Return of $k$-bonacci random walks

In this work, the probability of return for random walks on $\mathbb{Z}$, whose increment is given by the $k$-bonacci sequence, is determined. Also, the Hausorff, packing and box-counting dimensions of the set of these walks that return an infinite number of times to the origin are given. As an application, we study the return for tribonacci random walks to the first term of the tribonacci sequence.

math.PR↗

arXiv2019

A multifractal formalism for Hewitt-Stromberg measures

In the present work, we give a new {\it multifractal formalism} for which the classical multifractal formalism does not hold. We precisely introduce and study a multifractal formalism based on the Hewitt-Stromberg measures and that this formalism is completely parallel to Olsen's multifractal formalism which based on the Hausdorff and packing measures.

math.DS↗

arXiv2014

Hausdorff and packing spectra, large deviations, and free energy for branching random walks in $\R^d$

Consider an $\R^d$-valued branching random walk (BRW) on a supercritical Galton Watson tree. Without any assumption on the distribution of this BRW we compute, almost surely and simultaneously, the Hausdorff and packing dimensions of the level sets $E(K)$ of infinite branches in the boundary of the tree (endowed with its standard metric) along which the averages of the BRW have a given closed connected set of limit points $K$. This goes beyond multifractal analysis, which only considers those level sets when $K$ ranges in the set of singletons $\{α\}$, $α\in\R^d$. We also give a $0$-$\infty$ law for the Hausdorff and packing measures of the level sets $E(\{α\})$, and compute the free energy of the associated logarithmically correlated random energy model in full generality. Moreover, our results complete the previous works on multifractal analysis by including the levels $α$ which do not belong to the range of the gradient of the free energy. This covers in particular a situation until now badly understood, namely the case where a first order phase transition occurs. As a consequence of our study, we can also describe the whole singularity spectrum of Mandelbrot measures, as well as the associated free energy function (or $L^q$-spectrum), when a first order phase transition occurs.

math-ph↗