Related papers: The second minimum of Barnes-Wall lattices
Given a polarization of an even unimodular lattice and integer $k\ge 1$, we define a family of unimodular lattices $L(M,N,k)$. Of special interest are certain $L(M,N,3)$ of rank 72. Their minimum norms lie in $\{4, 6, 8\}$. Norms 4 and 6 do…
We consider a branching random walk in the non-boundary case where the additive martingale $W_n$ converges a.s. and in mean to some non-degenerate limit $W_\infty$. We first establish the joint tail distribution of $W_\infty$ and the global…
We prove that the very simple lattices which consist of a largest, a smallest and $2n$ pairwise incomparable elements where $n$ is a positive integer can be realized as the lattices of intermediate subfactors of finite index and finite…
We provide simple proofs of analogues for coverings numbers of lattices of several recently studied basic statements on the ranks of tensors. We highlight the differences and analogies between the proofs in both settings.
We classify the unimodular equivalence classes of inclusion-minimal polygons with a certain fixed lattice width. As a corollary, we find a sharp upper bound on the number of lattice points of these minimal polygons.
We obtain lower bound for the maximum distance between any three distinct points in an affine lattice which are close to a helix with small curvature and torsion.
In this paper, we present an improved continued fraction approximation of the Wallis ratio. This approximation is fast in comparison with the recently discovered asymptotic series. We also establish the double-side inequality related to…
We use GL(2) delta method to establish the Burgess bound.
In this article, we study the relation between lattice basis and successive minima and give an estimate for the measure-theoretical distribution of successive minima. As consequences, we also discuss some logarithm laws associated to higher…
We give an exponential improvement to the lower bound on diagonal Ramsey numbers for any fixed number of colors greater than two.
We provide new families of minimal codes in any characteristic. Also, an inductive construction of minimal codes is presented.
In this paper, we obtain two new lower bounds for the smallest singular value of nonsingular matrices which is better than the bound presented by zou \cite{zou2012lower}, Lin, Minghua and Xie, Mengyan \cite{lin2021some} under certain…
We improve the lower bound for the minimum number of colors for linear Alexander quandle colorings of a knot given in Theorem 1.2 of Colorings beyond Fox: The other linear Alexander quandles (Linear Algebra and its Applications, Vol. 548,…
We prove that, for any $B \subset \mathbb R$, the Cartesian product set $B \times B$ determines $\Omega(|B|^{2+c})$ distinct angles.
In this paper, we study a classical construction of lattices from number fields and obtain a series of new results about their minimum distance and other characteristics by introducing a new measure of algebraic numbers. In particular, we…
Knots have been considered to be useful models for simulating molecular chains such as DNA and proteins. One quantity that we are interested on molecular knots is the minimum number of monomers necessary to realize a knot. In this paper we…
We answer a question of Erd\H{o}s by showing that the least modulus of a distinct covering system of congruences is no larger than $10^{18}$.
Using linear projections one gets new inequalities for the successive minima of the lattice of sections of an hermitian line bundle on an arithmetic surface.
We prove minimal entropy rigidity for complete, finite volume manifolds locally isometric to a product of rank one symmetric spaces of dimension at least 3: the locally symmetric metric uniquely minimizes (normalized) entropy among all…
The goal of this expository note is to give a short, self-contained proof of nearly optimal lower bounds for the second largest eigenvalue of the adjacency matrix of regular graphs.