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Osama Khalil

Publications and source records attributed to Osama Khalil.

18 recordsLinked to original sources
arXiv2026

Strong spectral gap for geometrically finite hyperbolic manifolds

Let $\Gamma < G := \operatorname{SO}(d+1, 1)$ for $d \geq 1$ be a Zariski dense, geometrically finite, discrete subgroup with critical exponent strictly greater than $d/2$. We show that $L^2(\Gamma\backslash G)$ admits a strong spectral gap, confirming a conjecture of Mohammadi and Oh. This extends the spherical spectral gap on $L^2(\Gamma\backslash \mathbb{H}^{d+1}) \cong L^2(\Gamma\backslash G/\operatorname{SO}(d+1))$, which follows by the works of Lax-Phillips, Patterson, and Sullivan by different methods. As a consequence, we establish rates of decay of matrix coefficients, and of exponential mixing of the frame flow, that are explicitly determined by the size of the strong spectral gap.

math.DS↗
arXiv2025

Veech Surfaces and Expanding Twist Tori on Moduli Spaces of Abelian Differentials

Let $(M,\omega)$ be a translation surface such that every leaf of its horizontal foliation is either closed, or joins two zeros of $\omega$. Then, $M$ decomposes as a union of horizontal Euclidean cylinders. The $\textit{twist torus}$ of $(M,\omega)$, denoted $\mathbb{T}(\omega)$, consists of all translation surfaces obtained from $(M,\omega)$ by applying the horocycle flow independently to each of these cylinders. Let $g_t$ be the Teichm\"uller geodesic flow. We study the distribution of the expanding tori $g_t\cdot \mathbb{T}(\omega)$ on moduli spaces of translation surfaces in cases where $(M,\omega)$ is a $\textit{Veech surface}$. We provide sufficient criteria for these tori to become dense within the conjectured limiting locus $\mathcal{M} :=\overline{\mathrm{SL}_2(\mathbb{R})\cdot \mathbb{T}(\omega)}$ as $t\rightarrow \infty$. We also provide criteria guaranteeing a uniform lower bound on the mass a given open set $U\subset\mathcal{M}$ must receive with respect to any weak-$\ast$ limit of the uniform measures on $g_t\cdot \mathbb{T}(\omega)$ as $t\rightarrow\infty$. In particular, all such limits must be fully supported in $\mathcal{M}$ in such cases. Finally, we exhibit infinite families of well-known examples of Veech surfaces satisfying each of these results. A key feature of our results in comparison to previous work is that they do not require passage to subsequences.

math.DS↗
arXiv2025

$L^2$-Flattening of Self-similar Measures on Non-degenerate Curves

Let $\mu$ be a non-atomic self-similar measure on $\mathbb{R}$, and let $\nu$ be its pushforward to a non-degenerate curve in $\mathbb{R}^d, d\geq 1$. We show that for every $\epsilon>0$, there is $p>1$, so that $\left \lVert \hat{\nu} \right \rVert_{L^p(B(R))}^p = O_\epsilon(R^\epsilon)$ for all $R>1$, where $B(R)$ is the $R$-ball about the origin. As a corollary, we show that convolution with $\nu$ quantitatively improves $L^2$-dimension.

math.CA↗
arXiv2025

Development of a PPO-Reinforcement Learned Walking Tripedal Soft-Legged Robot using SOFA

Rigid robots were extensively researched, whereas soft robotics remains an underexplored field. Utilizing soft-legged robots in performing tasks as a replacement for human beings is an important stride to take, especially under harsh and hazardous conditions over rough terrain environments. For the demand to teach any robot how to behave in different scenarios, a real-time physical and visual simulation is essential. When it comes to soft robots specifically, a simulation framework is still an arduous problem that needs to be disclosed. Using the simulation open framework architecture (SOFA) is an advantageous step. However, neither SOFA's manual nor prior public SOFA projects show its maximum capabilities the users can reach. So, we resolved this by establishing customized settings and handling the framework components appropriately. Settling on perfect, fine-tuned SOFA parameters has stimulated our motivation towards implementing the state-of-the-art (SOTA) reinforcement learning (RL) method of proximal policy optimization (PPO). The final representation is a well-defined, ready-to-deploy walking, tripedal, soft-legged robot based on PPO-RL in a SOFA environment. Robot navigation performance is a key metric to be considered for measuring the success resolution. Although in the simulated soft robots case, an 82\% success rate in reaching a single goal is a groundbreaking output, we pushed the boundaries to further steps by evaluating the progress under assigning a sequence of goals. While trailing the platform steps, outperforming discovery has been observed with an accumulative squared error deviation of 19 mm. The full code is publicly available at \href{https://github.com/tarekshohdy/PPO_SOFA_Soft_Legged_Robot.git}{github.com/tarekshohdy/PPO$\textunderscore$SOFA$\textunderscore$Soft$\textunderscore$Legged$\textunderscore$ Robot.git}

cs.RO↗
arXiv2025

Automating Hot-Rolling: Designing an Integrated Mechatronics System for Enhanced Efficiency in Sheet Metal Production

The hot-rolling process is a critical stage in sheet metal production within the heavy steel industry. Traditionally, parameter adjustments such as sheet metal velocity and roll gap are performed manually, leading to inefficiencies and limited precision. This project introduces an integrated mechatronics system designed to automate the control of rolling speed and sheet metal thickness, enhancing efficiency, consistency, and quality. The proposed system consists of a pair of rolls applying compression loads, with a mechanism for gap control, suitable motors and sensors, and dynamic modeling to optimize performance. Through simulation and practical implementation strategies, we demonstrate the feasibility of automating the hot-rolling process. By integrating mechatronics, this solution aims to modernize sheet metal production, improve productivity, and enhance product quality in the steel industry.

eess.SY↗
arXiv2025

Measure rigidity and equidistribution for fractal carpets

Let $\theta$ be a Bernoulli measure which is stationary for a random walk generated by finitely many contracting rational affine dilations of $\mathbb{R}^d$, and let $\mathcal{K} = \mathrm{supp}(\theta)$ be the corresponding attractor. An example in dimension $d=1$ is the Hausdorff measure on Cantor's middle thirds set, and examples in higher dimensions include missing digits sets, Sierpi\'nski carpets and Menger sponges. Let $\nu$ denote the image of $\theta$ under the map $\mathcal{K} \to \mathrm{SL}_{d+1}(\mathbb{R})/\mathrm{SL}_{d+1}(\mathbb{Z})$ which sends $x$ to the lattice $\Lambda_x = \mathrm{span}_{\mathbb{Z}}(e_1,\ldots,e_d,e_{d+1} + (x,0))$. We prove equidistribution of the pushforward measures $a_{n*}\nu$ along any diverging sequence of diagonal matrices $(a_n)\subset\mathrm{SL}_{d+1}(\mathbb{R})$ that expand the first $d$ coordinates under a natural non-escape of mass condition. The latter condition is known to hold whenever $\theta$ is absolutely friendly. We also show that weighted badly approximable vectors and Dirichlet-improvable vectors (for arbitrary norm) form a subset of $\mathcal{K}$ of $\theta$-measure zero. The key ingredient is a measure classification theorem for the stationary measures of an associated random walk on an $S$-arithmetic space, introduced by the two first-named authors in earlier work. A new feature of this setting is that this random walk admits stationary measures which are not invariant.

math.DS↗
arXiv2024

Fourier Decay from $L^2$-Flattening

We develop a unified approach for establishing rates of decay for the Fourier transform of a wide class of dynamically defined measures. Among the key features of the method is the systematic use of the $L^2$-flattening theorem obtained in \cite{Khalil-Mixing}, coupled with non-concentration estimates for the derivatives of the underlying dynamical system. This method yields polylogarithmic Fourier decay for Diophantine self-similar measures, and polynomial decay for Patterson-Sullivan measures of convex cocompact hyperbolic manifolds, Gibbs measures associated to non-integrable $C^2$ conformal systems, as well as stationary measures for carpet-like non-conformal iterated function systems. Applications include essential spectral gaps on convex cocompact hyperbolic manifolds, fractal uncertainty principles, and equidistribution properties of typical vectors in fractal sets.

math.DS↗
arXiv2024

Polynomial Fourier Decay For Patterson-Sullivan Measures

We show that the Fourier transform of Patterson-Sullivan measures associated to convex cocompact groups of isometries of real hyperbolic space decays polynomially quickly at infinity. The proof is based on the $L^2$-flattening theorem obtained in prior work of the author, combined with a method based on dynamical self-similarity for ruling out the sparse set of potential frequencies where the Fourier transform can be large.

math.DS↗
arXiv2023

On the Space of Ergodic Measures for the Horocycle Flow on Strata of Abelian Differentials

We study the horocycle flow on the stratum of translation surfaces $\mathcal{H}(2)$. We show that there is a sequence of horocycle ergodic measures, each supported on a periodic horocycle orbit, which weakly converges to an invariant, but non-ergodic, measure by $\mathrm{SL}_2(\mathbb{R})$. As a consequence, we show that there are points in $\mathcal{H}(2)$ whose horocycle flow orbits do not equidistribute towards any invariant measure.

math.DS↗
arXiv2023

Exponential Mixing Via Additive Combinatorics

We prove that the geodesic flow on a geometrically finite locally symmetric space of negative curvature is exponentially mixing with respect to the Bowen-Margulis-Sullivan measure. The approach is based on constructing a suitable anisotropic Banach space on which the infinitesimal generator of the flow admits an essential spectral gap. A key step in the proof involves estimating certain oscillatory integrals against the Patterson-Sullivan measure. For this purpose, we prove a general result of independent interest asserting that the Fourier transform of measures on $\mathbb{R}^d$ that do not concentrate near proper affine hyperplanes enjoy polynomial decay outside of a sparse set of frequencies. As an intermediate step, we show that the $L^q$-dimension ($1<q\leq \infty$) of iterated self-convolutions of such measures tend towards that of the ambient space. Our analysis also yields that the Laplace transform of the correlation function of smooth observables extends meromorphically to the entire complex plane in the convex cocompact case and to a strip of explicit size beyond the imaginary axis in the case the manifold admits cusps.

math.DS↗
arXiv2022

Random Walks, Spectral Gaps, and Khintchine's Theorem on Fractals

This work addresses problems on simultaneous Diophantine approximation on fractals, motivated by a long standing problem of Mahler regarding Cantor's middle $1/3$ set. We obtain the first instances where a complete analogue of Khintchine's Theorem holds for fractal measures. Our results apply to fractals which are self-similar by a system of rational similarities of $\mathbb{R}^d$ (for any $d\geq 1$) and have sufficiently small Hausdorff co-dimension. A concrete example of such measures in the context of Mahler's problem is the Hausdorff measure on the "middle $1/5$ Cantor set"; i.e. the set of numbers whose base $5$ expansions miss a single digit. The key new ingredient is an effective equidistribution theorem for certain fractal measures on the homogeneous space $\mathcal{L}_{d+1}$ of unimodular lattices; a result of independent interest. The latter is established via a new technique involving the construction of $S$-arithmetic operators possessing a spectral gap and encoding the arithmetic structure of the maps generating the fractal. As a consequence of our methods, we show that spherical averages of certain random walks naturally associated to the fractal measures effectively equidistribute on $\mathcal{L}_{d+1}$.

math.DS↗
arXiv2021

Geodesic Planes in Geometrically Finite Manifolds

We study the problem of rigidity of closures of totally geodesic plane immersions in geometrically finite manifolds containing rank $1$ cusps. We show that the key notion of K-thick recurrence of horocycles fails generically in this setting. This property was introduced in the recent work of McMullen, Mohammadi and Oh. Nonetheless, in the setting of geometrically finite groups whose limit sets are circle packings, we derive 2 density criteria for non-closed geodesic plane immersions, and show that closed immersions give rise to surfaces with finitely generated fundamental groups. We also obtain results on the existence and isolation of proper closed immersions of elementary surfaces.

math.DS↗
arXiv2020

Bounded and Divergent Trajectories And Expanding Curves on Homogeneous Spaces

Suppose $g_t$ is a $1$-parameter $\mathrm{Ad}$-diagonalizable subgroup of a Lie group $G$ and $Γ< G$ is a lattice. We study the dimension of bounded and divergent orbits of $g_t$ emanating from a class of curves lying on leaves of the unstable foliation of $g_t$ on the homogeneous space $G/Γ$. We obtain sharp upper bounds on the Hausdorff dimension of divergent on average orbits and show that the set of bounded orbits is winning in the sense of Schmidt (and, hence, has full dimension). The class of curves we study is roughly characterized by being tangent to copies of $\mathrm{SL}(2,\mathbb{R})$ inside $G$, which are not contained in a proper parabolic subgroup of $G$. We describe applications of our results to problems in Diophantine approximation by number fields and intrinsic Diophantine approximation on spheres. Our methods also yield the following result for lines in the space of square systems of linear forms: suppose $φ(s) = sY + Z$ where $Y\in \mathrm{GL}(n,\mathbb{R})$ and $Z\in M_{n,n}(\mathbb{R})$. Then, the dimension of the set of points $s$ such that $φ(s)$ is singular is at most $1/2$ while badly approximable points have Hausdorff dimension equal to $1$.

math.DS↗
arXiv2020

Singular Vectors on Fractals and Projections of Self-similar Measures

Singular vectors are those for which the quality of rational approximations provided by Dirichlet's Theorem can be improved by arbitrarily small multiplicative constants. We provide an upper bound on the Hausdorff dimension of singular vectors lying on self-similar fractals in $\mathbb{R}^d$ satisfying the open set condition. The bound is in terms of quantities which are closely tied to Frostman exponents of projections of the Hausdorff measure supported on the fractal. Our bound is optimal in the sense that it agrees with the exact dimension of singular vectors obtained by Cheung and Chevallier when the fractal is trivial (i.e. has non-empty interior). As a corollary, we show that if the fractal is the product of $2$ copies of Cantor's middle thirds set or the attractor of a planar homogeneous irrational IFS, then the upper bound is $2/3$ the dimension of the fractal. This addresses the upper bound part of a question raised by Bugeaud, Cheung and Chevallier. We apply our method in the setting of translation flows on flat surfaces to show that the dimension of non-uniquely ergodic directions belonging to a fractal is at most $1/2$ the dimension of the fractal.

math.DS↗
arXiv2017

Exceptional directions for the Teichm\"{u}ller geodesic flow and Hausdorff dimension

We prove that for every flat surface $\omega$, the Hausdorff dimension of the set of directions in which Teichm\"{u}ller geodesics starting from $\omega$ exhibit a definite amount of deviation from the correct limit in Birkhoff's and Oseledets' Theorems is strictly less than $1$. This theorem extends a result by Chaika and Eskin where they proved that such sets have measure $0$. We also prove that the Hausdorff dimension of the directions in which Teichm\"{u}ller geodesics diverge on average in a stratum is bounded above by $1/2$, strengthening a classical result due to Masur. Moreover, we show that the Hausdorff codimension of the set of non-weakly mixing IETs with permutation $(d, d-1, \dots, 1)$, where $d$ is an odd number, is exactly $1/2$ and strengthen a result by Avila and Leguil.

math.DS↗
arXiv2017

Pointwise Equidistribution and Translates of Measures on Homogeneous Spaces

Let $(X,\mathfrak{B},μ)$ be a Borel probability space. Let $T_n: X\rightarrow X$ be a sequence of continuous transformations on $X$. Let $ν$ be a probability measure on $X$ such that $\frac{1}{N}\sum_{n=1}^N (T_n)_\ast ν\rightarrow μ$ in the weak-$\ast$ topology. Under general conditions, we show that for $ν$ almost every $x\in X$, the measures $\frac{1}{N}\sum_{n=1}^N δ_{T_n x}$ get equidistributed towards $μ$ if $N$ is restricted to a set of full upper density. We present applications of these results to translates of closed orbits of Lie groups on homogeneous spaces. As a corollary, we prove equidistribution of exponentially sparse orbits of the horocycle flow on quotients of $SL(2,\mathbb{R})$, starting from every point in almost every direction.

math.DS↗
arXiv2012

Viewpoint Invariant Object Detector

Object Detection is the task of identifying the existence of an object class instance and locating it within an image. Difficulties in handling high intra-class variations constitute major obstacles to achieving high performance on standard benchmark datasets (scale, viewpoint, lighting conditions and orientation variations provide good examples). Suggested model aims at providing more robustness to detecting objects suffering severe distortion due to < 60° viewpoint changes. In addition, several model computational bottlenecks have been resolved leading to a significant increase in the model performance (speed and space) without compromising the resulting accuracy. Finally, we produced two illustrative applications showing the potential of the object detection technology being deployed in real life applications; namely content-based image search and content-based video search.

cs.CV↗