On strong Sidon sets of integers
Suppose \(0<\alpha<1\). A set \(S\subset\mathbb{N}\) is called a \textit{Sidon} set if all sums \(a+b\) with \(a\le b\) in \(S\) are distinct. An equivalent formulation of this notion would be that for every \(x,y,z,w\in S\) with \(x<y\le z<w\) we have that \(|(x+w)-(y+z)|\ge 1\). This latter notion allows us to make define (as the authors of this paper did in an earlier work) an \textit{ \(\alpha\)-strong Sidon set} to be a set \(S\subset\mathbb{N}\) satisfying the property that for every \(x,y,z,w\in S\) with \(x<y\le z<w\) we have that \(|(x+w)-(y+z)|\ge w^{\alpha}\). A finite-set version is as follows: A subset \(S\subset [n]:=\{1,\ldots,n\}\) is called an \textit{\((n,\alpha)\)-strong Sidon set} if for every \(x,y,z,w\in S\) with \(x<y\le z<w\) we have that \(|(x+w)-(y+z)|\ge n^{\alpha}\). This raises the following natural extremal problem: Determine (at least asymptotically) \(F(n,\alpha)\), the maximum size of an \((n,\alpha)\)-strong Sidon subset of \([n]\). \\ Among the results in this paper, the dominant term for \(F(n,\alpha)\); more precisely, \(F(n,\alpha)=n^{(1-\alpha)/2}+\) lower order terms, though the asymptotics there are not exact. The authors also show that if \(S\subset\mathbb{N}\) is an \(\alpha\)-strong Sidon set, then there exists a constant \(c=c(\alpha)\) such that \(S(n)\le cn^{(1-\alpha)/2}\) for all sufficiently large \(n\) (Here, \(S(n)=|S\cap[n]|\)). The authors also build on an older result of Erdős for lower bounds on Sidon sets to obtain an analogous result for \(\alpha\)-strong Sidon subsets of \(\mathbb{N}\). \\ The last result of the paper considers the problem of finding large Sidon sets. More precisely, the question of whether there are Sidon sets \(S_{\varepsilon}\) in \(\mathbb{N}\) with \(S(n)\ge n^{1/2-\varepsilon}\) remains open, and the best known result is due to Ruzsa who constructed such sets with \(S(n)\ge n^{\sqrt{2}-1+o(1)}\). The third result of this paper proves a similar result for \(\alpha\)-strong Sidon sets, i.e., constructs \(\alpha\)-strong Sidon sets in \(\mathbb{N}\) with \(S(n)\ge n^{\frac{\sqrt{2}-1+o(1)}{1+32\sqrt{\alpha}}}\) for \(0\le \alpha\le 10^{-4}\).
scientific article; zbMATH DE number 7371419