Minimum number of additive tuples in groups of prime order (Q668091)

Summary: 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. We observe that an elegant argument of \textit{W. Samotij} and \textit{B. Sudakov} [Math. Proc. Camb. Philos. Soc. 160, No. 3, 495--512 (2016; Zbl 1371.11030)] 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 \textit{B. Bajnok} [Additive combinatorics. A menu of research problems. Boca Raton, FL: CRC Press (2018; Zbl 1415.11001)], when \(a_0=\dots=a_k=:a\) and \(A_0=\dots=A_k\), provided \(k\) is not equal 1 modulo \(p\). Finally, by applying basic Fourier analysis, we show for Bajnok's problem that if \(p\geq 13\) and \(a\in\{3,\dots,p-3\}\) are fixed while \(k\equiv 1\pmod p\) tends to infinity, then the extremal configuration alternates between at least two affine non-equivalent sets.