Finding large additive and multiplicative Sidon sets in sets of integers

Given h,g ∈ N h , g ∈ N , we write a set X ⊂ Z X ⊂ Z to be a Bₕ⁺[g] B h + [ g ] set if for any n ∈ Z n ∈ Z , the number of solutions to the additive equation n = x₁ + + xₕ n = x 1 + ⋯ + x h with x₁, , xₕ ∈ X x 1 , ⋯ , x h ∈ X is at most g , where we consider two such solutions to be the same if they differ only in the ordering of the summands. We define a multiplicative Bₕ×[g] B h × [ g ] set analogously. In this paper, we prove, amongst other results, that there exist absolute constants g ∈ N g ∈ N and δ >0 δ > 0 such that for any h ∈ N h ∈ N and for any finite set A of integers, the largest Bₕ⁺[g] B h + [ g ] set B inside A and the largest Bₕ×[g] B h × [ g ] set C inside A satisfy aligned max \ |B|, |C| \ ₕ |A|(1+ δ )/h. aligned max { | B | , | C | } ≫ h | A | ( 1 + δ ) / h . In fact, when $$h=2$$ h = 2 , we may set $$g = 31$$ g = 31 , and when h is sufficiently large, we may set $$g = 1$$ g = 1 and δ (log log h)1/2 - o(1) δ ≫ ( log log h ) 1 / 2 - o ( 1 ) . The former makes progress towards a recent conjecture of Klurman–Pohoata and quantitatively strengthens previous work of Shkredov.