A periodicity lemma in linear diophantine analysis

Abstract

Let g = ( g 1,…, g r ) ≥ 0 and h = ( h 1,…, h r ) ≥ 0, g ϱ , h ϱ ∈ J , be two vectors of nonnegative integers and let λ ϵ J , λ ≥ 0, λ ≡ 0 mod d, where d denotes g.c.d. ( g 1,…, g r ). Define Δ(λ)=Δ(λg,h):= min ∑ ϱ=1 r x ϱh ϱ:x ϱ⩾0,x ϱ∈J, ∑ ϱ=1 ϱ x ϱg ϱ=λ It is shown in this paper that Λ(λ) is periodic in λ with constant jump. If i ϵ {1,…, r} is such that det g i h i g ϱ h ϱ ⩾ (ϱ1,…r) then Δ(λ)+g iΔ(λ)+h i holds true for all sufficiently large λ, λ ≡ 0 mod d.