Minimum number of additive tuples in groups of prime order

Katherine Staden (University of Oxford)

Frank Adams 1,

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

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