Discrete & Computational Geometry

, Volume 39, Issue 1–3, pp 59–66

Siegel’s Lemma and Sum-Distinct Sets

Article

Abstract

Let L(x)=a1x1+a2x2+⋅⋅⋅+anxn, n≥2, be a linear form with integer coefficients a1,a2,…,an 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 a1,a2,…,an. The main result of this paper asserts that there exist linearly independent vectors x1,…,xn−1∈ℤn such that L(xi)=0, i=1,…,n−1, and
$$\|{\mathbf{x}}_{1}\|\cdots\|{\mathbf{x}}_{n-1}\|<\frac{\|{\mathbf{a}}\|}{\sigma_{n}},$$
where a=(a1,a2,…,an) and
$$\sigma_{n}=\frac{2}{\pi}\int_{0}^{\infty}\left(\frac{\sin t}{t}\right)^{n}\,dt.$$

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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