Elisabeth Nossek
In this article we prove that integral lattices with minimum <= 7 (or <= 9) whose set of minimal vectors form spherical 9-designs (or 11-designs respectively) are extremal, even and unimodular. We furthermore show that there does not exist an integral lattice with minimum <=11 which yields a 13-design.
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We study the connection between the theory of spherical designs and the question of extrema of the height function of lattices. More precisely, we show that a full-rank n-dimensional Euclidean lattice, all layers of which hold a spherical 2-design, realises a stationary point for the height function, which is defined as the first derivative at 0 of the spectral zeta function of the associated flat torus. Moreover, in order to find out the lattices for which this 2-design property holds, a strategy is described which makes use of theta functions with spherical coefficients, viewed as elements o…
We study lattice points in d-dimensional spheres, and count their number in thin spherical segments. We found an upper bound depending only on the radius of the sphere and opening angle of the segment. To obtain this bound we slice the segment by hyperplanes of rational direction, and then cover an arbitrary segment with one having rational direction. Diophantine approximation can be used to obtain the best rational direction possible.
We set up a connection between the theory of spherical designs and the question of minima of Epstein's zeta function. More precisely, we prove that a Euclidean lattice, all layers of which hold a 4-design, achieves a local minimum of the Epstein's zeta function, at least at any real s>n/2. We deduce from this a new proof of Sarnak and Strömbergsson's theorem asserting that the root lattices D4 and E8, as well as the Leech lattice, achieve a strict local minimum of the Epstein's zeta function at any s>0. Furthermore, our criterion enables us to extend their theorem to all the so-called extremal…
Boris Venkov passed away on November 10 2011 just 5 days before his 77th birthday. This article gives a short survey of the mathematical work of Boris Venkov in this direction.
Lattice point counting statistics for 3-dimensional shrinking Cygan-Korányi spherical shells
本文研究三维收缩 Cygan-Korányi 球壳中的整数格点计数误差，证明归一化误差在间隙宽度趋于零且正则时具有由矩确定的极限分布，并刻画高斯与非高斯情形。
In this paper we study spherical equidistribution on the space of (translates of) adelic lattices, which we apply to understand the fine-scale statistics of the directions in the set of shifted primitive lattice points. We also apply our results to the distribution of the free path lengths in the Boltzmann--Grad limit for point sets such as (possibly non-rational) translates of the lattice points all of whose coordinates are squarefree. Besides the equidistribution results for translates of expanding horospheres, a key ingredient is a probabilistic argument which allows us to tackle the techni…
We consider lattices generated by finite Abelian groups. We prove that such a lattice is strongly eutactic, which means the normalized minimal vectors of the lattice form a spherical 2-design, if and only if the group is of odd order or if it is a power of the group of order 2. This result also yields a criterion for the appropriately normalized minimal vectors to constitute a uniform normalized tight frame. Further, our result combined with a recent theorem of R. Bacher produces (via the classical Voronoi criterion) a new infinite family of extreme lattices. Additionally, we investigate the s…
The density Hypothesis for irreducible lattices in $\mathrm{PSL}_2(\mathbb{R})^d$
本文证明了 $\mathrm{PSL}_2(\mathbb{R})^d$ 中不可约均匀格子的同余子群的密度假设，将球面情形推广至非温非球面表示，并给出关于水平和谱参数的一致界。
Spherical designs and modular forms of the $D_4$ lattice
本文证明$D_4$格壳层构成对跖球面$\{10,4,2\}$-设计、2-壳层为紧设计且24点情形唯一，并将其应用于格的唯一性与模形式Fourier系数。
Spherical statistics and phase transitions in high-dimensional lattices
该研究分析了高维随机格薄壳中点的球面统计，发现了多个相变阈值，并证明单个格在概率趋于一的情况下满足多种统计估计，且通过Lean形式化验证了主要结论。
Restriction estimates for toral eigenfunctions and lattice points in spherical regions
本文通过切片与打包方法结合离散球面乘子逼近，建立了环面本征函数的新L^2限制估计，在某些情形下达到最优，从而证明了Huang-Zhang关于大余维光滑子流形的猜想，并为Bourgain-Rudnick猜想提供了新进展。
The Voronoi Spherical CDF for Lattices and Linear Codes: New Bounds for Quantization and Coding
本文提出Voronoi球CDF用于格和线性码，推导出新的非渐近上界，改进了量化和编码的性能分析。
Let X be a tight t-design of dimension n for one of the open cases t=5 or t=7. An investigation of the lattice generated by X using arithmetic theory of quadratic forms allows to exclude infinitely many values for n.
In 1947, Lehmer conjectured that the Ramanujan's tau function $τ(m)$ never vanishes for all positive integers $m$, where $τ(m)$ is the $m$-th Fourier coefficient of the cusp form $Δ_{24}$ of weight 12. The theory of spherical $t$-design is closely related to Lehmer's conjecture because it is shown, by Venkov, de la Harpe, and Pache, that $τ(m)=0$ is equivalent to the fact that the shell of norm $2m$ of the $E_{8}$-lattice is a spherical 8-design. So, Lehmer's conjecture is reformulated in terms of spherical $t$-design. Lehmer's conjecture is difficult to prove, and still remains open. However,…
We study energy minimization for pair potentials among periodic sets in Euclidean spaces. We derive some sufficient conditions under which a point lattice locally minimizes the energy associated to a large class of potential functions. This allows in particular to prove a local version of Cohn and Kumar's conjecture that $\mathsf{A}_2$, $\mathsf{D}_4$, $\mathsf{E}_8$ and the Leech lattice are globally universally optimal, regarding energy minimization, and among periodic sets of fixed point density.
We consider the question of how approximations satisfying Dirichlet's theorem spiral around vectors in $\mathbb{R}^d$. We give pointwise almost everywhere results (using only the Birkhoff ergodic theorem on the space of lattices). In addition, we show that for $\textit{every}$ unimodular lattice, on average, the directions of approximates spiral in a uniformly distributed fashion on the $d-1$ dimensional unit sphere. For this second result, we adapt a very recent proof of Marklof and Strömbergsson \cite{MS3} to show a spherical average result for Siegel transforms on $\operatorname{SL}_{d+1}(\…
In 1947, Lehmer conjectured that the Ramanujan $τ$-function $τ(m)$ never vanishes for all positive integers $m$, where the $τ(m)$ are the Fourier coefficients of the cusp form $Δ_{24}$ of weight 12. Lehmer verified the conjecture in 1947 for $m<214928639999$. In 1973, Serre verified up to $m<10^{15}$, and in 1999, Jordan and Kelly for $m<22689242781695999$. The theory of spherical $t$-design, and in particular those which are the shells of Euclidean lattices, is closely related to the theory of modular forms, as first shown by Venkov in 1984. In particular, Ramanujan's $τ$-function gives the c…
Equality in a Reverse Minkowski Shell Bound for Integral Lattices via Spherical Designs
本文通过球面设计理论，分类了Regev-Stephens-Davidowitz关于整格壳层计数不等式的等式情形，证明了仅当格为Z^n或E_8时等式成立。
There are many analogies between codes, lattices, and vertex operator algebras. For example, extremal objects are good examples of combinatorial, spherical, and conformal designs. In this study, we investigated these objects from the aspect of design theory.
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