On a question of Sárkozy and Sós for bilinear forms

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