Linear Forms and Complementing Sets of Integers
نویسنده
چکیده
Let φ(x1, . . . , xh, y) = u1x1 + · · · + uhxh + vy be a linear form with nonzero integer coefficients u1, . . . , uh, v. Let A = (A1, . . . , Ah) 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 φ and the sets A and B as follows: R (φ) A,B (n) = card ({(a1, . . . , ah, b) ∈ A1 × · · · × Ah × B : φ(a1, . . . , ah, 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 {φ(a1, . . . , ah, 0) : (a1, . . . , ah) ∈ A1 × · · · × Ah}. 1. Representation functions for linear forms Let h ≥ 1 and let ψ(x1, . . . , xh) = u1x1 + · · ·+ uhxh be a linear form with nonzero integer coefficients u1, . . . , uh. Let A = (A1, . . . , Ah) be an h-tuple of sets of integers. The image of ψ with respect to A is the set ψ(A) = {ψ(a1, . . . , ah) : (a1, . . . , ah) ∈ A1 × · · · ×Ah} . Then ψ(A) 6= ∅ if and only if Ai 6= ∅ for all i = 1, . . . , h. For ψ(A) 6= ∅, we define the diameter of A with respect to ψ by D (ψ) A = diam(ψ(A)) = sup(ψ(A)) − inf(ψ(A)). We have D (ψ) A > 0 if and only if |Ai| > 1 for some i. For every integer n, we define the representation function associated to ψ by R (ψ) A (n) = card ({(a1, . . . , ah) ∈ A1 × · · · ×Ah : ψ(a1, . . . , ah) = n}) . Then n ∈ ψ(A) if and only if R (ψ) A (n) > 0. Let l ≥ 1 and let ω(y1, . . . , yl) = v1y1 + · · ·+ vlyl be another linear form with nonzero integer coefficients v1, . . . , vl. Consider the linear form φ(x1, . . . , xh, y1, . . . , yl) = ψ(x1, . . . , xh) + ω(y1, . . . , yl). Date: February 2, 2008. 2000 Mathematics Subject Classification. 11B34, 11B13, 11B75,11A67,11D04,11D72.
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