A set A ⊆ N is (k, `)-sum-free, for k, ` ∈ N, k > `, if it contains no solutions to the equation x1 + · · ·+xk = y1 + · · ·+y`. Let ρ = ρ(k− `) be the smallest natural number not dividing k − `, and let r = rn, 0 ≤ r < ρ, be such that r ≡ n (mod ρ). The main result of this note says that if (k − `)/` is small in terms of ρ, then the number of (k, `)-sum… (More)

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