Siegels’s lemma is sharp for almost all linear systems

The well-known Siegel Lemma gives an upper bound $cU^{m/(n−m)}$ for the size of the smallest non-zero 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.