Lp Linear discrepancy of totally unimodular matrices

Abstract

Let p ∈ [1, ∞[ and c p = max a ∈ [0, 1] ((1 − a) a p + a(1 − a) p ) 1/ p . We prove that the known upper bound lindisc p ( A) ⩽ c p for the L p linear discrepancy of a totally unimodular matrix A is asymptotically sharp, i.e., sup A lindisc p ( A ) = c p . We estimate c p = p p + 1 1 p + 1 1 / p ( 1 + ε p ) for some ε p ∈ [0, 2 − p+2 ], hence c p = 1 - ln p p ( 1 + o ( 1 ) ) . We also show that an improvement for smaller matrices as in the case of L ∞ linear discrepancy cannot be expected. For any p ∈ N we give a totally unimodular ( p + 1) × p matrix having L p linear discrepancy greater than p p + 1 1 p + 1 1 / p .