Subset Sums Avoiding Quadratic Nonresidues

It is a well-known problem to give an estimate for the largest clique of the Paley-graph, i.e. , to give an estimate for |A| if A ⊂ Fp (p ≡ 1 (mod 4)) is such that A−A = {a−a′ |a, a′ ∈ A} avoids the set of quadratic nonresidues. In this paper we will study a much simpler problem namely when A− A is substituted by the set FS(A) = { ∑ εaa | εa = 0 or 1 and ∑ εa > 0}. In other words we will estimate the maximal cardinality of A ⊂ Fp if FS(A) avoids the set of quadratic nonresidues. We will show that this problem is strongly related to the problem of the estimation of the least quadratic nonresidue. If n(p) denotes the least quadratic nonresidue then the set {1, 2, . . . , [n(p)]} satis es the conditions, this already gives a lower bound for the maximal value of |A|. Later we will prove that the maximal value of |A| is Ω(log log p). On the other hand we will prove that |A| = O(n(p) log p). The proof is based on the fact that if t is a quadratic nonresidue then FS(A) ∩ t · FS(A) = ∅ or {0} where by de nition t · B = {tb | b ∈ B}. We will show that if t is small than |FS(A)| is much greater than |A|. In the next section we will study the case when t = n(p) = 2. In the third part we will prove the upper bound |A| = O(n(p) log p). In the last part we will show that the maximal value of |A| is Ω(log log p).