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

We prove that for pairwise co-prime numbers $k_1,\dots,k_d \geq 2$ there does not exist any infinite set of positive integers $A$ such that the representation 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 a particular case of our main theorem, which poses a further step towards answering a question of S\'ark\"ozy and S\'os and widely extends a previous result of Cilleruelo and Ru\'e for bivariate linear forms.
Comment: Added clarifications regarding the particular notion of limit used in the first part of the paper. 11 pages