Siegel's lemma is sharp for almost all linear systems

The well‐known Siegel Lemma gives an upper bound c U m / ( n − m )for the size of the smallest non‐zero integral solution of a linear system of m ⩾ 1 equations in n > m unknowns whose coefficients are integers of absolute value at most U ⩾ 1 ; here c = c ( m , n ) ⩾ 1 . In this paper, we show that a better upper boundU m / ( n − m ) / B is relatively rare for large B ⩾ 1 ; for example, there are θ = θ ( m , n ) > 0 andc ′ = c ′ ( m , n )such that this happens for at mostc ′ U m n / B θout of the roughly( 2 U ) m npossible such systems.