On additive complements. II

On minimal additive complements of integers

“…For any integer l with K ≤ l < L, there exist e ∈ E and w ∈ W such that l = e + w. As w ≥ min{0, y − }, it follows from (2) that e = l − w < L − min{0, y − } ≤ N 0 , and so e ∈ E − i j for some integer i j . Suppose that w ∈ Y (0) ∪ Y (1) . Then we have y − ≤ w ≤ y + and K − y + ≤ e = l − w < L − y − .…”

“…For any integer e ∈ E with K ≤ e < L, there exists a w ∈ W such that e + w = e ′ + w ′ for any e ′ = e, e ′ ∈ E and w ′ ∈ W . Now we may assume that w ∈ Y (1) . It is enough to prove that e + w ≡…”

“…Theorem 1. Let W be defined in (1). If there exists a minimal complement to W , then there exists C ⊆ {0, 1, .…”

“…Subcase 2.2. w ∈ Y(1) . Since {c + y} mod m ⊆ C + X m mod m for any y ∈ Y(1) and d ≡ c (mod m), w ∈ Y (1) , it follows that {n} mod m = {d + w} mod m ⊆ C + X m mod m. Hence there exist a c ′ ∈ D with c ′ (mod m) ∈ C and x ∈ W with x mod m ∈ X m such that n ≡ c ′ + x (mod m). We choose a sufficiently large integer k such that c ′ − km ∈ D, c ′ − km = d and x + km ∈ W .…”

Asymptotic complements in the integers

“…See [Erd54], [Erd57] etc. Some more recent results on asymptotic complements can be found in [Wol96,FC10,CF11,FC14] etc.…”