# Inverse problems for linear forms over finite sets of integers

Melvyn B. Nathanson

Arxiv ID: 0708.2304•Last updated: 1/6/2021

Let f(x_1,x_2,...,x_m) = u_1x_1+u_2 x_2+... + u_mx_m be a linear form with
positive integer coefficients, and let N_f(k) = min{|f(A)| : A \subseteq 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 (u_1, u_2,...,u_m) of positive integers is called
complete if {\sum_{j\in J} u_j : J \subseteq {1,2,..,m}} = {0,1,2,..., U},
where $U = \sum_{j=1}^m u_j.$ It is proved that if f is an m-ary linear form
whose coefficient sequence (u_1,...,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.

#### PaperStudio AI Chat

I'm your research assistant! Ask me anything about this paper.