Linear Discrepancy of Totally Unimodular Matrices

Let p [1, [ and cp = maxa[0,1]((1 - a)ap + a(1 - a)p)1/p. We prove that the known upper bound lindiscp(A) cp for the Lp linear discrepancy of a totally unimodular matrix A is asymptotically sharp, i.e., sup A lindiscp(A) = cp. We estimate cp = p p+1 1 p+1 1/p (1+p) for some p [0, 2-p+2], hence cp = 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)

• Type: Journal
• Year: 2004
• Where: COMBINATORICA
• Authors: Benjamin Doerr
• Keywords: COMBINATORICA 2004, Linear Discrepancy, Lp Linear Discrepancy, Unimodular Matrix