Yael Eisenberg | Searcharxiv

arXiv2023

On the Shortest Lattice Vector vs. the Shortest Basis

Given an arbitrary basis for a mathematical lattice, to find a ``good" basis for it is one of the classic and important algorithmic problems. In this note, we give a new and simpler proof of a theorem by Regavim (arXiv:2106.03183): we construct a 18-dimensional lattice that does not have a basis that satisfies the following two properties simultaneously: 1. The basis includes the shortest non-zero lattice vector. 2. The basis is shortest, that is, minimizes the longest basis vector (alternatively: the sum or the sum-of-squares of the basis vectors). The vectors' length can be measured in any $\ell^q$ norm, for $q\in \mathbb{N}_+$ (albeit, via another lattice, of a somewhat larger dimension).

math.MG↗

arXiv2022

A Tight Reverse Minkowski Inequality for the Epstein Zeta Function

We prove that if $\mathcal{L} \subset \mathbb{R}^n$ is a lattice such that $\det(\mathcal{L}') \geq 1$ for all sublattices $\mathcal{L}' \subseteq \mathcal{L}$, then \[ \sum_{\substack{\mathbf{y}\in\mathcal{L}\\\mathbf{y}\neq\mathbf0}} (\|\mathbf{y}\|^2+q)^{-s} \leq \sum_{\substack{\mathbf{z} \in \mathbb{Z}^n\\\mathbf{z}\neq\mathbf{0}}} (\|\mathbf{z}\|^2+q)^{-s} \] for all $s > n/2$ and all $0 \leq q \leq (2s-n)/(n+2)$, with equality if and only if $\mathcal{L}$ is isomorphic to $\mathbb{Z}^n$.

math.MG↗