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Naser T. Sardari

Publications and source records attributed to Naser T. Sardari.

12 recordsLinked to original sources
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↗
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↗
arXiv2020

Asymptotic trace formula for the Hecke operators

Given integers $m$, $n$ and $k$, we give an explicit formula with an optimal error term (with square root cancelation) for the Petersson trace formula involving the $m$-th and $n$-th Fourier coefficients of an orthonormal basis of $S_k(N)^*$ (the weight $k$ newforms with fixed square-free level $N$) provided that $|4 π\sqrt{mn}- k|=o(k^{\frac{1}{3}})$. Moreover, we establish an explicit formula with a power saving error term for the trace of the Hecke operator $\mathcal{T}_n^*$ on $S_k(N)^*$ averaged over $k$ in a short interval. By bounding the second moment of the trace of $\mathcal{T}_{n}$ over a larger interval, we show that the trace of $\mathcal{T}_n$ is unusually large in the range $|4 π\sqrt{n}- k| = o(n^{\frac{1}{6}})$. As an application, for any fixed prime $p$ with $\gcd(p,N)=1$, we show that there exists a sequence $\{k_n\}$ of weights such that the error term of Weyl's law for $\mathcal{T}_p$ is unusually large and violates the prediction of arithmetic quantum chaos. In particular, this generalizes the result of Gamburd, Jakobson and Sarnak~\cite[Theorem 1.4]{Gamburd} with an improved exponent.

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.

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arXiv2019

Bounds on the multiplicity of the Hecke eigenvalues

Fix an integer $N$ and a prime $p\nmid N$ where $p\geq 5$. We show that the number of newforms $f$ (up to a scalar multiple) of level $N$ and even weight $k$ such that $\mathcal{T}_p(f)=0$ is bounded independently of $k$, where $\mathcal{T}_p(f)$ is the Hecke operator.

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arXiv2019

The least prime number represented by a binary quadratic form

Let $D<0$ be a fundamental discriminant and $h(D)$ be the class number of $\mathbb{Q}(\sqrt{D})$. Let $R(X,D)$ be the number of classes of the binary quadratic forms of discriminant $D$ which represent a prime number in the interval $[X,2X]$. Moreover, assume that $π_{D}(X)$ is the number of primes, which split in $\mathbb{Q}(\sqrt{D})$ with norm in the interval $[X,2X].$ We prove that $$ \Big(\frac{π_D(X)}{π(X)}\Big)^2 \ll \frac{R(X,D)}{h(D)}\Big(1+\frac{h(D)}{π(X)}\Big), $$ where $π(X)$ is the number of primes in the interval $[X,2X]$ and the implicit constant in $\ll$ is independent of $D$ and $X$.

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arXiv2019

The Siegel variance formula for quadratic forms

We introduce a smooth variance sum associated to a pair of positive definite symmetric integral matrices $A_{m\times m}$ and $B_{n\times n}$, where $m\geq n$. By using the oscillator representation, we give a formula for this variance sum in terms of a smooth sum over the square of a functional evaluated on the $B$-th Fourier coefficients of the vector valued holomorphic Siegel modular forms which are Hecke eigenforms and obtained by the theta transfer from $O_{A_{m\times m}}$. By using the Ramanujan bound on the Fourier coefficients of the holomorphic cusp forms, we give a sharp upper bound on this variance when $n=1$. As applications, we prove a cutoff phenomenon for the probability that a unimodular lattice of dimension $m$ represents a given even number. This gives an optimal upper bound on the sphere packing density of almost all even unimodular lattices. Furthermore, we generalize the result of Bourgain, Rudnick and Sarnak~\cite{Bourgain}, and also give an optimal bound on the diophantine exponent of the $p$-integral points on any positive definite $d$-dimensional quadric, where $d\geq 3$. This improves the best known bounds due to Ghosh, Gorodnik and Nevo~\cite{GGN} into an optimal bound.

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arXiv2018

The diophantine exponent of the $\mathbb{Z}/q\mathbb{Z}$ points of $S^{d-2}\subset S^d$

Assume a polynomial-time algorithm for factoring integers, Conjecture~\ref{conj}, $d\geq 3,$ and $q$ and $p$ are prime numbers, where $p\leq q^A$ for some $A>0$. We develop a polynomial-time algorithm in $\log(q)$ that lifts every $\mathbb{Z}/q\mathbb{Z}$ point of $S^{d-2}\subset S^{d}$ to a $\mathbb{Z}[1/p]$ point of $S^d$ with the minimum height. We implement our algorithm for $d=3 \text{ and }4$. Based on our numerical results, we formulate a conjecture which can be checked in polynomial-time and gives the optimal bound on the diophantine exponent of the $\mathbb{Z}/q\mathbb{Z}$ points of $S^{d-2}\subset S^d$.

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arXiv2018

The least prime ideal in a given ideal class

Let $K$ be a number field with the discriminant $D_K$ and the class number $h_{K}$, which has bounded degree over $\mathbb{Q}$. By assuming GRH, we prove that every ideal class of $K$ contains a prime ideal with norm less than $h_{K}^2\log(D_K)^{2}$ and also all but $o(h_K)$ of them have a prime ideal with norm less than $h_{K}\log(D_K)^{2+ε}$. For imaginary quadratic fields $K=\mathbb{Q}(\sqrt{D})$, by assuming Conjecture~\ref{piarcor} (a weak version of the pair correlation conjecure), we improve our bounds by removing a factor of $\log(D)$ from our bounds and show that these bounds are optimal.

math.NT↗
arXiv2018

The Cramér conjecture holds with a positive probability

We prove that a positive proportion of the intervals of any fixed scalar multiple of $\log(X)$ in the dyadic interval $[X,2X]$ contain a prime number. We also show that a positive proportion of the congruence classes modulo $q$ contain a prime number smaller than any fixed scalar multiple of $φ(q)\log(q).$

math.NT↗
arXiv2017

Quantum Chaos on random Cayley graphs of ${\rm SL}_2[\mathbb{Z}/p\mathbb{Z}]$

We investigate the statistical behavior of the eigenvalues and diameter of random Cayley graphs of ${\rm SL}_2[\mathbb{Z}/p\mathbb{Z}]$ %and the Symmetric group $S_n$ as the prime number $p$ goes to infinity. We prove a density theorem for the number of exceptional eigenvalues of random Cayley graphs i.e. the eigenvalues with absolute value bigger than the optimal spectral bound. Our numerical results suggest that random Cayley graphs of ${\rm SL}_2[\mathbb{Z}/p\mathbb{Z}]$ and the explicit LPS Ramanujan projective graphs of $\mathbb{P}^1(\mathbb{Z}/p\mathbb{Z})$ have optimal spectral gap and diameter as the prime number $p$ goes to infinity.

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