A uniform stability principle for dual lattices

Abstract

We prove a highly uniform stability or ``almost-near'' theorem for dual lattices of
lattices $L \subseteq \Bbb R^n$. More precisely, we show that, for a vector $x$ from the linear span of a lattice $L \subseteq \Bbb R^n$, subject to $\lambda_1(L) \ge \lambda > 0$, to be $\varepsilon$-close to some vector from the dual lattice $L^\star$ of $L$, it is enough that the inner products $u\,x$ are $\delta$-close (with $\delta < 1/3$) to some integers for all vectors $u \in L$ satisfying $\| u \| \le r$, where $r > 0$ depends on $n$, $\lambda$, $\delta$ and $\varepsilon$, only. This generalizes an analogous result previously proved by M. Mačaj and the second author for integral vector lattices. The proof is nonconstructive, using the ultraproduct construction and a slight portion of nonstandard analysis.