## Minimum number of additive tuples in groups of prime order

#### Katherine Staden (University of Oxford)

For a prime number $$p$$ and a sequence of integers $$a_0,\dots,a_k\in \{0,1,\dots,p\}$$, let $$s(a_0,\dots,a_k)$$ be the minimum number of $$(k+1)$$-tuples $$(x_0,\dots,x_k)\in A_0\times\dots\times A_k$$ with $$x_0=x_1+\dots + x_k$$, over subsets $$A_0,\dots,A_k\subseteq\mathbb{Z}{p}$$ of sizes $$a_0,\dots,a_k$$ respectively. I will describe how an elegant argument of Samotij and Sudakov can be extended to show that there exists an extremal configuration with all sets $$A_i$$ being intervals of appropriate length. The same conclusion also holds for the related problem, posed by Bajnok, when $$a_0=\dots=a_k=:a$$ and $$A_0=\dots=A_k$$, provided $$k$$ is not equal 1 modulo $$p$$. In the remaining case, some basic Fourier analysis reveals different behaviour. Joint work with Ostap Chervak and Oleg Pikhurko