Inverse problems for linear forms over finite sets of integers

Details Inverse problems for linear forms over finite sets of integers Inverse problems for linear forms over finite sets of integers

2007-08-17

arxiv.org/abs/0708.2304v2

Mathematics

Number Theory

12 pages; minor corrections

Scientific paper

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.

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