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 $\delta 0$ depends on $n$, $\lambda$, $\delta$ and $\varepsilon$, only. This generalizes an analogous result previously proved by M. Macaj and the second author for integral vector lattices. The proof is nonconstructive, using the ultraproduct construction and a slight portion of nonstandard analysis.
Highlights
We prove a highly uniform stability or almost-near theorem for dual lattices of lattices L ⊆ Rn
The study of stability of the homomorphy property with respect to the compact-open topology was commenced by the second of the present authors in [22], [23], and [24]
In the present paper we will prove the stability result for dual lattices stated in the Abstract and formulated in a more detailed way in Theorem 5.2, as well as some closely related results
Summary
We assume some basic knowledge of lattices or, more generally, of “geometry of numbers”. The Minkowski successive minima of L (with respect to the unit ball B) are defined by λk(L) = inf{λ ∈ R : λ > 0, rank(L ∩ λB) ≥ k}. The covering radius of L is defined by μ(L) = inf{r ∈ R : r > 0, span(L) ⊆ L + rB} In all these cases the infima are minima. Vk), with k < m, be an ordered k-tuple of linearly independent vectors from L which can be extended to a basis of L. Vm) of a lattice L is Minkowski reduced if, for each k ≤ m, vk is the shortest vector from L such that the k-tuple Introducing an orthonormal basis in the linear subspace span(L) and replacing any vector x ∈ span(L) by its coordinates with respect to it, they can be readily generalized as follows. Λk(L) λm−k+1 L ≤ m for each k ≤ m, and λ1(L) μ L
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