Infinite Sidon-type sets for zero-sum linear forms

Let h ≥ 2 h ≥ 2 , and let b = (b₁, ,bₕ)∈ Zʰ b = ( b 1 , ⋯ , b h ) ∈ Z h be a zero-sum vector with nonzero coordinates. For a set A=₁<a₂<⋯ \⊆ N A = { a 1 < a 2 < ⋯ } ⊆ N , let rA,b(n) r A , b ( n ) denote the number of h -tuples (x₁,… ,xₕ) ( x 1 , … , x h ) of pairwise distinct elements of A satisfying b₁x₁+⋯ +bₕxₕ=n b 1 x 1 + ⋯ + b h x h = n . We study density restrictions on sets A for which these representation counts remain small, obtaining analogues of the classical density theorem for infinite Sidon sets. In the case b = (c₁,-c₁, ,cₖ,-cₖ) b = ( c 1 , - c 1 , ⋯ , c k , - c k ) , we prove that if A(x)/(x/log x)1/2k→ ∞ A ( x ) /