Minimum number of additive tuples in groups of prime order

Journal article

For a prime number p and a sequence of integers a0,…,ak∈{0,1,…,p}, let s(a0,…,ak) be the minimum number of (k+1)-tuples (x0,…,xk)∈A0×⋯×Ak with x0=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, when a0=⋯=ak=:a and A0=⋯=Ak, provided k is not equal 1 modulo p. Finally, by applying basic Fourier analysis, we show for Bajnok's problem that if p⩾13 and a∈{3,…,p−3} are fixed while k≡1(modp)tends to infinity, then the extremal configuration alternates between at least two affine non-equivalent sets.

Authors: Chervak, O; Pikhurko, O; Staden, K

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Electronic Journal of Combinatorics
• Journal: Electronic Journal of Combinations
• Volume: 26
• Issue: 1
• Article number: P1.30
• Publication date: 2019-02-22
• Acceptance date: 2019-01-30
• ISSN: 1077-8926