Abstract
AbstractGiven $$h,g \in {\mathbb {N}}$$ h , g ∈ N , we write a set $$X \subset {\mathbb {Z}}$$ X ⊂ Z to be a $$B_{h}^{+}[g]$$ B h + [ g ] set if for any $$n \in {\mathbb {Z}}$$ n ∈ Z , the number of solutions to the additive equation $$n = x_1 + \dots + x_h$$ n = x 1 + ⋯ + x h with $$x_1, \dots , x_h \in X$$ 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 $$B_{h}^{\times }[g]$$ B h × [ g ] set analogously. In this paper, we prove, amongst other results, that there exist absolute constants $$g \in {\mathbb {N}}$$ g ∈ N and $$\delta >0$$ δ > 0 such that for any $$h \in {\mathbb {N}}$$ h ∈ N and for any finite set A of integers, the largest $$B_{h}^{+}[g]$$ B h + [ g ] set B inside A and the largest $$B_{h}^{\times }[g]$$ B h × [ g ] set C inside A satisfy $$\begin{aligned} \max \{ |B|, |C| \} \gg _{h} |A|^{(1+ \delta )/h }. \end{aligned}$$ 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 $$\delta \gg (\log \log h)^{1/2 - o(1)}$$ 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.
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