Additive and multiplicative Sidon sets (Q5898295)

Let \(A\) be a subset of \(\{1,2,\dots,n\}\) or \(\mathbb Z_n\). \(A\) is said to be an additive Sidon set if for every \(s\) the equation \(x+y=s\) has at most one solution with \(x,y\in A\). \(A\) is called a multiplicative Sidon set if for every \(s\) the equation \(x\cdot y=s\) has at most one solution with \(x,y\in A\). If \(A\) has both an additive and a multiplicative Sidon set we shall call it a doubly Sidon set. Denote by \(f(n)\), \(g(n)\), \(h(n)\) the size of the maximal additive Sidon set, the multiplicative Sidon set or the doubly Sidon set, respectively, contained in \(\mathbb Z_n\). Similarly let \(F(n)\), \(G(n)\), \(H(n)\) denote the analogous functions by considering as Sidon set the subset of \(\{1,2,\dots,n\}\). The main results are: \(\limsup \frac{h(n)}{\sqrt{n}}=\frac12\) and \(\liminf \frac{H(n)}{\sqrt{n}}=\frac12\). Some related questions are also discussed.