Linear forms and complementing sets of integers

Details Linear forms and complementing sets of integers Linear forms and complementing sets of integers

2008-01-02

arxiv.org/abs/0801.0001v1

Mathematics

Number Theory

10 pages

Scientific paper

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) \in 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 will be bounded in terms of the diameter of the finite set {phi(a_1,...,a_h,0): (a_1,..., a_h) \in A_1 x ... x A_h}.

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