Inverse Problems for Linear forms Over Finite Sets of Integers

Let f(x₁, x₂, . . . , x_m) = u₁x₁ + u₂x₂ + · · · + umxm be a linear form with positive integer coefficients, and let N_f (k) = min{|f(A)| : A ⊆ Z and |A| = k}. A minimizing k-set for f is a set A such that |A| = k and |f(A)| = N_f (k). A finite sequence (u₁, u₂, . . . , u_m) of positive integers is called complete if ∑_j∈J u_j : J ⊆ {1, 2, . . . , m}} = {0, 1, 2, . . . , U}, where U =∑^m_j =1 u_j. It is proved that if f is an m-ary linear form whose coefficient sequence (u₁, . . . , u_m) is complete, then N_f (k) = Uk − U + 1 and the minimizing k-sets are precisely the arithmetic progressions of length k. Other extremal results on linear forms over finite sets of integers are obtained.