A uniform stability principle for dual lattices

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 $lambda_1(L) ge lambda u003e 0$, 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 $delta 0$ depends on $n$, $lambda$, $delta$ and $varepsilon$, only. This generalizes an analogous result proved for integral vector lattices in cite{MZ}. The proof is nonconstructive, using the ultraproduct construction and a slight portion of nonstandard analysis.