A periodicity lemma in linear diophantine analysis

Let g = (g₁,…,g_r) ≥ 0 and h = (h₁,…,h_r) ≥ 0, g_?, h_? ∈ $\text{J}$, be two vectors of nonnegative integers and let λ ? $\text{J}$, λ ≥ 0, λ ≡ 0 mod d, where d denotes g.c.d. (g₁,…,g_r). Define

$\text{Δ(λ)=Δ(λg,h):=}\text{min}\text{∑}\text{?=1}\text{r}\text{x}{}_{\mathrm{?}}\text{h}{}_{\mathrm{?}}\text{:x}{}_{\mathrm{?}}\text{?0,x}{}_{\mathrm{?}}\text{∈J,}\text{∑}\text{?=1}\text{?}\text{x}{}_{\mathrm{?}}\text{g}{}_{\mathrm{?}}\text{=λ}$

It is shown in this paper that Λ(λ) is periodic in λ with constant jump. If i? {1,…,r} is such that

$\text{det}\text{g}{}_{\mathrm{i}}\text{h}{}_{\mathrm{i}}\text{g}{}_{\mathrm{?}}\text{h}{}_{\mathrm{?}}\text{? (?1,…r)}$

then

$\text{Δ(λ)+g}{}_{\mathrm{i}}\text{Δ(λ)+h}{}_{\mathrm{i}}$

holds true for all sufficiently large λ, λ ≡ 0 mod d.