2017/01/10 by Martin Vodička, Vodička, Martin, Pavol Zlatoš +1
Mathematics · #03H05 (Secondary) #11H06 (Primary) #11H31 #11H60 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:03H05 #msc:11H06 #msc:11H31 #msc:11H60
paper · pdf · doi:10.48550/arxiv.1701.02548
arxiv created 2018/08/15 · arxiv updated 2018/08/16
We prove a highly uniform stability or "almost-near" theorem for dual lattices of lattices L ⊆ \Bbb Rn. More precisely, we show that, for a vector x from the linear span of a lattice L ⊆ \Bbb Rn, subject to λ1(L) ≥ λ> 0, to be ε-close to some vector from the dual lattice L' of L, it is enough that the inner products u x are δ-close (with δ< 1/3) to some integers for all vectors u ∈ L satisfying ‖ u ‖ ≤ r, where r > 0 depends on n, λ, δ and ε, 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.