Abstract
Let L(x)=a 1 x 1+a 2 x 2+⋅⋅⋅+a n x n , n≥2, be a linear form with integer coefficients a 1,a 2,…,a n which are not all zero. A basic problem is to determine nonzero integer vectors x such that L(x)=0, and the maximum norm ‖x‖ is relatively small compared with the size of the coefficients a 1,a 2,…,a n . The main result of this paper asserts that there exist linearly independent vectors x 1,…,x n−1∈ℤn such that L(x i )=0, i=1,…,n−1, and
where a=(a 1,a 2,…,a n ) and
This result also implies a new lower bound on the greatest element of a sum-distinct set of positive integers (Erdös–Moser problem). The main tools are the Minkowski theorem on successive minima and the Busemann theorem from convex geometry.
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The work was partially supported by FWF Austrian Science Fund, Project M821-N12.
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Aliev, I. Siegel’s Lemma and Sum-Distinct Sets. Discrete Comput Geom 39, 59–66 (2008). https://doi.org/10.1007/s00454-008-9059-9
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DOI: https://doi.org/10.1007/s00454-008-9059-9