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On a question of Sárkozy and Sós for bilinear forms

We prove that, if 2 ⩽ k1 ⩽ k2, then there is no infinite sequence 𝒜 of positive integers such that the representation function r(n) = #{(a, a′): n = k1a + k2a′, a, a′ ∈ 𝒜} is constant for n large enough. This result completes the previous work of Dirac and Moser for the special case k1 = 1 and answers a question posed by Sárkozy and Sós.