A uniform stability principle for dual lattices

Abstract

We prove a highly uniform stability or "almost-near" theorem for dual lattices of lattices $L\subseteq {\mathbb{R}}^n$. More precisely, we show that, for a vector $x$ from the linear span of a lattice $L\subseteq {\mathbb{R}}^n$, subject to ${\lambda }_1(L)\ge \lambda >0$, to be $\epsilon$-close to some vector from the dual lattice $L^{\prime }$ of $L$, it is enough that the inner products $ux$ are $\delta$-close (with $\delta <1/3$) to some integers for all vectors $u\in L$ satisfying $\Vert u\Vert \le r$, where $r>0$ depends on $n$, $\lambda$, $\delta$ and $\epsilon$, only. This generalizes an earlier analogous result proved for integral vector lattices by M. Mačaj and the second author. The proof is nonconstructive, using the ultraproduct construction and a slight portion of nonstandard analysis.