Linear forms and complementing sets of integers [article]

Melvyn B. Nathanson
2008 arXiv   pre-print
Let phi(x_1,...,x_h,y) = u_1x_1 + ... + u_hx_h+vy be a linear form with nonzero integer coefficients u_1,..., u_h, v. Let A = (A_1,..., A_h) be an h-tuple of finite sets of integers and let B be an infinite set of integers. Define the representation function associated to the form phi and the sets A and B as follows: R^(phi)_A,B(n) = card((a_1,..., a_h,b) ∈ A_1 x ... x A_h x B: phi(a_1, ..., a_h,b) = n). If this representation function is constant, then the set B is periodic and the period of B
more » ... will be bounded in terms of the diameter of the finite set phi(a_1,...,a_h,0): (a_1,..., a_h) ∈ A_1 x ... x A_h.
arXiv:0801.0001v1 fatcat:4s6jhwumkjaw3e3cmp7lwrj54a