Finding large additive and multiplicative Sidon sets in sets of integers

Given h,g is an element of N, we write a set X subset of Z to be a B-h(+)[g] set if for any n is an element of Z, the number of solutions to the additive equation n=x(1)+& ctdot;+x(h) with x(1),& mldr;,x(h )is an element of 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(x)[g] set analogously. In this paper, we prove, amongst other results, that there exist absolute constants g is an element of N and delta>0 such that for any h is an element of N and for any finite set A of integers, the largest B-h(+)[g] set B inside A and the largest B-h(x)[g] set C inside A satisfy max{|B|,|C|}>>(h)|A|((1+delta)/h). In fact, when h=2, we may set g=31, and when h is sufficiently large, we may set g=1 and delta >>(loglogh)(1/2-o(1)). The former makes progress towards a recent conjecture of Klurman--Pohoata and quantitatively strengthens previous work of Shkredov.