Baker, Roger and Masser, David. (2019) Siegels’s lemma is sharp for almost all linear systems. Preprints Fachbereich Mathematik, 2019 (12).

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Abstract
The wellknown Siegel Lemma gives an upper bound $cU^{m/(n−m)}$ for the size of the smallest nonzero integral solution of a linear system of $m \ge 1$ equations in $n > m$ unknowns whose coefficients are integers of absolute value at most $U \ge 1$; here $c = c(m, n) \ge 1$. In this paper we show that a better upper bound $U^{m/(n−m)}/B$ is relatively rare for large $B \ge 1$; for example there are $\theta = \theta(m,n) > 0$ and $c′ = c′(m,n)$ such that this happens for at most $c′U^{mn}/B^\theta$ out of the roughly $(2U)^{mn}$ possible such systems.
Faculties and Departments:  05 Faculty of Science > Departement Mathematik und Informatik > Ehemalige Einheiten Mathematik & Informatik > Zahlentheorie (Masser) 12 Special Collections > Preprints Fachbereich Mathematik 

UniBasel Contributors:  Masser, David 
Item Type:  Preprint 
Publisher:  Universität Basel 
Language:  English 
Last Modified:  16 Oct 2019 10:37 
Deposited On:  16 Oct 2019 10:37 
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