Minimum number of additive tuples in groups of prime order

2017/10/05 by Chervak, Ostap, Pikhurko, Oleg, Staden, Katherine · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1710.01936

Abstract

For a prime number p and a sequence of integers a₀,…,a_k∈ \0,1,…,p\, let s(a₀,…,a_k) be the minimum number of (k+1)-tuples (x₀,…,x_k)∈ A₀×…× A_k with x₀=x₁+… + x_k, over subsets A₀,…,A_k⊆ℤ_p of sizes a₀,…,a_k respectively. An elegant argument of Lev (independently rediscovered by Samotij and Sudakov) shows that there exists an extremal configuration with all sets A_i being intervals of appropriate length, and that the same conclusion also holds for the related problem, reposed by Bajnok, when a₀=…=a_k=:a and A₀=…=A_k, provided k is not equal 1 modulo p. 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\pmod p tends to infinity, then the extremal configuration alternates between at least two affine non-equivalent sets.

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