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We prove that for pairwise co-prime numbers $k_1,\dots,k_d \geq 2$ there doesnot exist any infinite set of positive integers $A$ such that therepresentation function $r_A (n) = \{ (a_1, \dots, a_d) \in A^d : k_1 a_1 +\dots + k_d a_d = n \}$ becomes constant for $n$ large enough. This result is aparticular case of our main theorem, which poses a further step towardsanswering a question of S\'ark\"ozy and S\'os and widely extends a previousresult of Cilleruelo and Ru\'e for bivariate linear forms.

On a problem of S\'ark\"ozy and S\'os for multivariate linear forms