The Helly bound for singular sums (Q1598849)

A set \(S\subseteq \{0,1,2,\dots,n-1\}\) is called singular if \(x-y\) is singular, that is not coprime to \(n\), for all \(x,y\in S\). Let \(S_2(n)\) be the maximum value of \(|S+S|\) over all singular \(S\subseteq \{0,1,2,\dots,n-1\}\). Suppose \(n\) has \(d\) prime factors, the smallest of which is \(p\). If \(d\leq 2\), then \(S_2(n)=n/p\). In this paper the language of graph theory and properties of Helly families are used to prove that in general \(S_2(n)\geq n-\phi(n)\), where \(\phi\) denotes the Euler totient function. This bound is shown to be tight when \(d=3\) and it is conjectured to be tight for \(d=4\) as well. For \(d\geq 5\) it is claimed that this bound can always be improved. An extension of \(S_2\) to finite abelian groups is also explored.