Complementary Sets of Finite Sets

Subsets 𝒜, 𝒮 of an additive group G are complementary if 𝒜 + 𝒮 = G. When 𝒜 is of finite cardinality ∣𝒜∣, and G is ℤ or ℝ, we give sufficient conditions for the existence of a complementary set 𝒮 with “density” not much larger than 1/∣𝒜∣.