arXiv subjects

Masoud Zargar

Publications and source records attributed to Masoud Zargar.

9 recordsLinked to original sources

arXiv2024

Stiefel manifolds and upper bounds for spherical codes and packings

We improve upper bounds on sphere packing densities and sizes of spherical codes in high dimensions. In particular, we prove that the maximal sphere packing densities $\delta_n$ in $\mathbb{R}^n$ satisfy \[\delta_n\leq \frac{1+o(1)}{e}\cdot \delta^{\text{KL}}_{n}\] for large $n$, where $\delta^{\text{KL}}_{n}$ is the best bound on $\delta_n$ obtained essentially by Kabatyanskii and Levenshtein from the 1970s with improvements over the years. We also obtain the same improvement factor for the maximal size $M(n,\theta)$ of $\theta$-spherical codes in $S^{n-1}$: for angles $0<\theta<\theta'\leq\frac{\pi}{2}$, \[M(n,\theta)\leq \frac{1+o(1)}{e}\cdot \frac{M_{\text{Lev}}(n-1,\theta')}{\mu_n(\theta,\theta')}\] for large $n$, where $\mu_n(\theta,\theta')$ is the mass of the spherical cap in the unit sphere $S^{n-1}$ of radius $\frac{\sin(\theta/2)}{\sin(\theta'/2)}$, and $M_{\text{Lev}}(n-1,\theta')$ is Levenshtein's upper bound on $M(n-1,\theta')$ when applying the Delsarte linear programming method to Levenshtein's optimal polynomials. In fact, we prove that there are no analytic losses in our arguments and that the constant $\frac{1}{e}=0.367...$ is optimal for the class of functions considered. Our results also show that the improvement factor does not depend on the special angle $\theta^*=62.997...^{\circ}$, explaining the numerics in arXiv:2001.00185. In the spherical codes case, the above inequality improves the Kabatyanskii--Levenshtein bound by a factor of $0.2304...$ on geometric average. Along the way, we construct a general class of functions using Stiefel manifolds for which we prove general results and study the improvement factors obtained from them in various settings.and study the improvement factors obtained from them in various settings.

math.MG↗

arXiv2023

New upper bounds for spherical codes and packings

We improve the previously best known upper bounds on the sizes of $θ$-spherical codes for every $θ<θ^*\approx 62.997^{\circ}$ at least by a factor of $0.4325$, in sufficiently high dimensions. Furthermore, for sphere packing densities in dimensions $n\geq 2000$ we have an improvement at least by a factor of $0.4325+\frac{51}{n}$. Our method also breaks many non-numerical sphere packing density bounds in smaller dimensions. This is the first such improvement for each dimension since the work of Kabatyanskii and Levenshtein~\cite{KL} and its later improvement by Levenshtein~\cite{Leven79}. Novelties of this paper include the analysis of triple correlations, usage of the concentration of mass in high dimensions, and the study of the spacings between the roots of Jacobi polynomials.

math.MG↗

arXiv2023

The union-closed sets conjecture for non-uniform distributions

The union-closed sets conjecture, attributed to Péter Frankl from 1979, states that for any non-empty finite union-closed family of finite sets not consisting of only the empty set, there is an element that is in at least half of the sets in the family. We prove a version of Frankl's conjecture for families distributed according to any one of infinitely many distributions. As a corollary, in the intersection-closed reformulation of Frankl's conjecture, we obtain that it is true for families distributed according to any one of infinitely many Maxwell--Boltzmann distributions with inverse temperatures bounded below by a positive universal constant. Frankl's original conjecture corresponds to zero inverse temperature.

math.CO↗

arXiv2022

Optimal strong approximation for quadrics over $\mathbb{F}_q[t]$

Suppose $q$ is a fixed odd prime power, $F(\vec{x})$ is a non-degenerate quadratic form over $\mathbb{F}_q[t]$ of discriminant $Δ$ in $d\geq 5$ variables $\vec{x}$, and $f,g\in\mathbb{F}_q[t]$, $\boldsymbolλ\in\mathbb{F}_q[t]^d$. We show that whenever $\text{deg} f\geq (4+\varepsilon)\text{deg} g+O_{\varepsilon,F}(1)$, $\gcd(Δ^{\infty},fg)=O(1)$, and the necessary local conditions are satisfied, we have a solution $\vec{x}\in\mathbb{F}_q[t]^d$ to $F(\vec{x})=f$ such that $\vec{x}\equiv\boldsymbolλ\bmod g$. For $d=4$, we show that the same conclusion holds if we instead have $\text{deg} f\geq (6+\varepsilon)\text{deg} g+O_{\varepsilon,F}(1)$. This gives us a new proof (independent of the Ramanujan conjecture over function fields proved by Drinfeld) that the diameter of any $k$-regular Morgenstern Ramanujan graphs $G$ is at most $(2+\varepsilon)\log_{k-1}|G|+O_{\varepsilon}(1)$. In contrast to the $d=4$ case, our result is optimal for $d\geq 5$. Our main new contributions are a stationary phase theorem over function fields for bounding oscillatory integrals, and a notion of anisotropic cones to circumvent isotropic phenomena in the function field setting.

math.NT↗

arXiv2022

Random flat bundles and equidistribution

Each signature $\underline{\lambda}(n)=(\lambda_1(n),\dots,\lambda_n(n))$, where $\lambda_1(n)\geq\dots\geq\lambda_n(n)$ are integers, gives an irreducible representation $\pi_{\underline{\lambda}(n)}:U(n)\rightarrow\text{GL}(V_{\underline{\lambda}(n)})$ of the unitary group $U(n)$. Suppose $X$ is a finite-area cusped hyperbolic surface, $\chi$ is a random surface representation in $\text{Hom}(\pi_1(X),U(n))$ equipped with a Haar unitary probability measure, and $(\underline{\lambda}(n))_{n=1}^{\infty}$ is a sequence of signatures. Let $|\underline{\lambda}(n)|:=\sum_i|\lambda_i(n)|$. We show that there is an absolute constant $c>0$ such that if $0\neq |\underline{\lambda}(n)|\leq c\frac{\log n}{\log\log n}$ for sufficiently large $n$, then the Laplacians $\Delta_{\chi,\underline{\lambda}(n)}$ acting on sections of the flat unitary bundles associated to the surface representations \[\pi_1(X)\xrightarrow{\chi} U(n)\xrightarrow{\pi_{\underline{\lambda}(n)}}\text{GL}(V_{\underline{\lambda}(n)})\] have the property that for every $\varepsilon>0$ \[\mathbb{P}\left[\chi:\inf\text{Spec}(\Delta_{\chi,\underline{\lambda}(n)})\geq\frac{1}{4}-\varepsilon\right]\xrightarrow{n\rightarrow\infty}1,\] where $\text{Spec}(\Delta_{\chi,\underline{\lambda}(n)})$ is the spectrum of $\Delta_{\chi,\underline{\lambda}(n)}$. A special case of this is that flat unitary bundles associated to $\chi:\pi_1(X)\rightarrow U(n)$ asymptotically almost surely as $n\rightarrow\infty$ have least eigenvalue at least $\frac{1}{4}-\varepsilon$, irrespective of the spectral gap of $X$ itself. This is proved using the Hide--Magee method. Using the spectral theorem above and proving a probabilistic prime geodesic theorem, we also obtain a probabilistic equidistribution theorem for the images under $\chi$ of geodesics of lengths dependent on the rank $n$.

math.NT↗

arXiv2020

Ramanujan graphs and exponential sums over function fields

We prove that $q+1$-regular Morgenstern Ramanujan graphs $X^{q,g}$ (depending on $g\in\mathbb{F}_q[t]$) have diameter at most $\left(\frac{4}{3}+\varepsilon\right)\log_{q}|X^{q,g}|+O_{\varepsilon}(1)$ (at least for odd $q$ and irreducible $g$) provided that a twisted Linnik-Selberg conjecture over $\mathbb{F}_q(t)$ is true. This would break the 30 year-old upper bound of $2\log_{q}|X^{q,g}|+O(1)$, a consequence of a well-known upper bound on the diameter of regular Ramanujan graphs proved by Lubotzky, Phillips, and Sarnak using the Ramanujan bound on Fourier coefficients of modular forms. We also unconditionally construct infinite families of Ramanujan graphs that prove that $\frac{4}{3}$ cannot be improved.

math.NT↗

arXiv2019

Riemannian structures and point-counting

Suppose $(X_n)$ is a sequence of positive-dimensional smooth projective complete intersections over $\mathbb{F}_q$ with dimensions bounded from above and with characteristic zero lifts $(\tilde{X}_n)$ to smooth projective geometrically connected varieties. Suppose each complex variety $\tilde{X}^{an}_n$ has (underlying real manifold equipped with) a Riemannian metric $g_n$ of sectional curvature at least $-κ_n^2$, $κ_n\geq 0$, and diameter at most $D_n$. In this note, we show that if \[\lim_{n\rightarrow\infty}\min\{d:X_n(\mathbb{F}_{q^d})\neq\emptyset\}=+\infty,\] then $κ_nD_n\rightarrow+\infty$. We deduce this theorem by proving a more general theorem estimating the number of points over finite fields of the above varieties in terms of the sectional curvature and diameter of Riemannian structures on the analytification of characteristic zero lifts. We also prove a version of this estimate for non-projective varieties equipped with complete metrics of non-negative sectional curvature. Finally, we prove a characteristic zero analogue of the above estimate by relating fixed-points of algebraic endomorphisms of smooth projective complex complete intersections to Riemannian structures on them.

math.AG↗

arXiv2019

Integration of Voevodsky motives

In this paper, we construct four different theories of integration, two that are for Voevodsky motives, one for mixed $\ell$-adic sheaves, and a fourth theory of integration for rational mixed Hodge structures. We then show that they circumvent some of the complications of classical motivic integration, leading to new arithmetic and geometric results concerning K-equivalent $k$-varieties. For example, in addition to recovering known results regarding K-equivalent smooth projective complex varieties, we show that K-equivalent smooth projective $\mathbb{F}_q$-varieties have isomorphic rational $\ell$-adic Galois representations (up to semisimplification), and so also the same zeta functions (the equality of zeta functions is true even without projectivity). This is an arithmetic result inaccessible to classical motivic integration. This paper also gives more evidence for a conjecture of Chin-Lung Wang suggesting the equivalence of integral motives of K-equivalent smooth projective varieties. Furthermore, we connect our theory of integration of rational Voevodsky motives to the existence of motivic $t$-structures for geometric Voevodsky motives; we show that the existence of a motivic t-structure implies that K-equivalent smooth projective varieties have equivalent rational (Chow) motives. We also connect this to a conjecture of Orlov concerning bounded derived categories of coherent sheaves. This makes progress on showing that all cohomology theories should agree for K-equivalent smooth projective varieties (at least rationally and for suitable base fields).

math.AG↗

arXiv2017

Comparison of Stable Homotopy Categories and a Generalized Suslin-Voevodsky Theorem

Let $k$ be an algebraically closed field of exponential characteristic $p$. Given any prime $\ell\neq p$, we construct a stable étale realization functor $$\underline{\text{Ét}}_{\ell}:\text{Spt}(k)\rightarrow \text{Pro}(\text{Spt})^{H\mathbb{Z}/\ell}$$ from the stable $\infty$-category of motivic $\mathbb{P}^1$-spectra over $k$ to the stable $\infty$-category of $(H\mathbb{Z}/\ell)^*$-local pro-spectra (see section 3 for definition). This is induced by the étale topological realization functor á la Friedlander. The constant presheaf functor naturally induces the functor \[\text{SH}[1/p]\rightarrow\text{SH}(k)[1/p],\] where $k$ and $p$ are as above and $\text{SH}$ and $\text{SH}(k)$ are the classical and motivic stable homotopy categories, respectively. We use the stable étale realization functor to show that this functor is fully faithful. Furthermore, we conclude with a homotopy theoretic generalization of the étale version of the Suslin-Voevodsky theorem.

math.AG↗