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For a prime number p and a sequence of integers a0, . . . , ak∈ {0,1, . . . , p}, lets (a0, . . . , ak) be the minimum number of (k+ 1)-tuples (x0, . . . , xk) ∈A0×···×Akwithx0=x1+···+xk, over subsets a0, . . . , Ak⊆Zp of sizes a0, . . . , ak respectively. We observe that an elegant argument of Samotij and Sudakov can be extended to show that there exists an extremal configuration with all sets Ai being intervals of appropriate length. The same conclusion also holds for the related problem ,posed by Bajnok, whena0=···=ak=:aandA0=···=Ak, provided k is not equal 1 modulop. Finally, by applying basic Fourier analysis, we show for Bajnok’s problem that if pu003e13 and a∈ {3, . . . , p−3}are fixed whilek≡1 (modp) tends to infinity, then the extremal configuration alternates between at least two affine non-equivalent sets.

Minimum Number of Additive Tuples in Groups of Prime Order