2024/05/07 by Sophie Huczynska, Huczynska, Sophie, Jonathan Jedwab +3 · 1 citation
Engineering · Mathematics · #05A15 #05E15 #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Rings, Modules, and Algebras #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2405.04638
openalex publication_date 2024/05/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let p be an odd prime. For nontrivial proper subsets A,B of ℤp of cardinality s,t, respectively, we count the number r(A,B,B) of additive triples, namely elements of the form (a, b, a+b) in A × B × B. For given s,t, what is the spectrum of possible values for r(A,B,B)? In the special case A=B, the additive triple is called a Schur triple. Various authors have given bounds on the number r(A,A,A) of Schur triples, and shown that the lower and upper bound can each be attained by a set A that is an interval of s consecutive elements of ℤp. However, there are values of p,s for which not every value between the lower and upper bounds is attainable. We consider here the general case where A,B can be distinct. We use Pollard's generalization of the Cauchy-Davenport Theorem to derive bounds on the number r(A,B,B) of additive triples. In contrast to the case A=B, we show that every value of r(A,B,B) from the lower bound to the upper bound is attainable: each such value can be attained when B is an interval of t consecutive elements of ℤp.