Restricted sums of cardinality \(1+p\) in a vector space over \(\mathbb{F}_p\) (Q5937932)

Let \(V\) be an abelian group and \(A, B\) be subsets of \(V\). The restricted sumset \(A +' B\) is the set of sums \(a+b\) of distinct elements of \(A\) and \(B\) (\(a \neq b\)). Here \(V\) is set to be any vector space over the finite field \(\mathbb{F}_p\) with \(p\) elements, \(p\) an odd prime, and \(A\) and \(B\) are subsets of cardinality \(r\) and \(s\) respectively. In an earlier paper [\textit{S. Eliahou} and \textit{M. Kervaire}, J. Number Theory 71, 12-39 (1998; Zbl 0935.11003)], the authors determined a number-theoretic function \(\gamma_p(r,s)\) such that \(|A +' B|\geq \gamma_p(r,s)\) and also proved that this lower bound is sharp when \((r,s)\) is not a so-called ``special pair'' (a condition on the \(p\)-adic expansions of \(r\) and \(s\)). In the case when \((r,s)\) is a special pair, they showed that the sharp lower bound is either \(\gamma_p(r,s)\) or \(\gamma_p(r,s) + 1\), but the problem of determining when each case occurs remains unsolved. NEWLINENEWLINENEWLINEThe simplest example of a special pair is \((r,s) = (1+p,1+p)\). The main result of the current paper is to prove for any vector space \(V\) over \(\mathbb{F}_p\), of dimension at least 2, and for \(p \geq 5\), that the sharp lower bound is given by \(\gamma_p(r,s) + 1\), which in this case is \(2p\).