Siegel's lemma is sharp for almost all linear systems
Roger C. Baker, David Masser · Bulletin of the London Mathematical Society · 2019
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.