Abstract:
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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). |