Additive triples in groups of odd prime order

Abstract: For a subset $A$ of an additive group $G$, a Schur triple in $A$ is a triple of the form $(a,b,a+b) \in A^3$. Denote by $r(A)$ the number of Schur triples of $A$; the behaviour of $r(A)$ as $A$ ranges over subsets of a group $G$ has been studied by various authors. When $r(A)=0$, $A$ is sum-free. The question of minimum and maximum $r(A)$ for $A$ of fixed size in $\mathbb{Z}_p$ was resolved by Huczynska, Mullen and Yucas (2009) and independently by Samotij and Sudakov (2016). Several generalisations of the Schur triple problem have received attention. In this talk, I will present recent work (with Jonathan Jedwab and Laura Johnson) on the generalisation to triples $(a,b,a+b) \in A \times B \times B$, where $A,B \subseteq \mathbb{Z}_p$. Denote by $r(A,B,B)$ the number of triples of this form; we obtain a precise description of its full spectrum of values and show constructively that each value in this spectrum can be realised when $B$ is an interval of consecutive elements in $\mathbb{Z}_p$.

Mathematics

Audience: researchers in the topic

Combinatorial and additive number theory (CANT 2025)