We prove that for pairwise co-prime numbers k1,...,kd = 2 there does not exist
any infinite set of positive integers A such that the representation function rA(n) =
#{(a1,...,ad) ¿ Ad : k1a1 + ... + kdad = 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 (Bull. of the London Math.
Soc...
We prove that for pairwise co-prime numbers k1,...,kd = 2 there does not exist
any infinite set of positive integers A such that the representation function rA(n) =
#{(a1,...,ad) ¿ Ad : k1a1 + ... + kdad = 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 (Bull. of the London Math.
Society 2009).
Citation
Rue, J., Spiegel, C. On a problem of Sárközy and Sós for multivariate linear forms. "Electronic notes in discrete mathematics", 1 Juliol 2018, vol. 68, núm. July 2018, p. 101-106.