Binary linear forms over finite sets of integers
Abstract
Let A be a finite set of integers. For a polynomial f(x1,...,xn) with integer coefficients, let f(A) = f(a1,...,an) : a1,...,an ∈ A. In this paper it is proved that for every pair of normalized binary linear forms f(x,y)=u1x+v1y and g(x,y)=u2x+v2y with integral coefficients, there exist arbitrarily large finite sets of integers A and B such that |f(A)| > |g(A)| and |f(B)| < |g(B)|.
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