Siegel's lemma is sharp for almost all linear systems. Issue 5 (27th August 2019) - Baker, Roger; Masser, David
Abstract: 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 bound U m / ( n − m ) / B is relatively rare for large B ⩾ 1 ; for example, there are θ = θ ( m, n ) > 0 and c ′ = c ′ ( m, n ) such that this happens for at most c ′ U m n / B θ out of the roughly ( 2 U ) m n possible such systems.