LOWER BOUNDS FOR DISCREPANCY

On lower bounds for the L2-discrepancy

The L2-discrepancy measures the irregularity of the distribution of a finite point set. In this note, we prove lower bounds for the L2-discrepancy of arbitrary N-point sets. Our main focus is on the two-dimensional case. Asymptotic upper and lower estimates of the L2-discrepancy in dimension 2 are well known, and are of the sharp order √ logN . Nevertheless, the gap in the constants between the...

Lower Bounds for the Discrepancy of Inversive Congruential Pseudorandom Numbers

The inversive congruential method is a uniform pseudorandom number generator which was introduced recently. For a prime modulus p the discrepancy D of k-tuples of successive pseudorandom numbers generated by this method determines the statistical independence properties of these pseudorandom numbers. It was shown earlier by the author that Dik) = 0(p~l,2(\oëp)k) for2<k<p. Here it is proved that...

Probabilistic Lower Bounds for the Discrepancy of Latin Hypercube Samples

We provide probabilistic lower bounds for the star discrepancy of Latin hypercube samples. These bounds are sharp in the sense that they match the recent probabilistic upper bounds for the star discrepancy of Latin hypercube samples proved in [M. Gnewuch, N. Hebbinghaus. Discrepancy bounds for a class of negatively dependent random points including Latin hypercube samples. Preprint 2016.]. Toge...

Some Lower Bounds for Estrada Index

A Discrepancy-Based Proof of Razborov’s Quantum Lower Bounds

In a breakthrough result, Razborov (2003) gave optimal lower bounds on the quantum communication complexity Q1/3( f ) of every function f (x, y) = D(|x ∧ y|), where D : {0, 1, . . . , n} → {0, 1}. Namely, he showed that Q1/3( f ) = Ω ( `1(D) + √ n `0(D) ) , where `0(D), `1(D) ∈ {0, 1, . . . , dn/2e} are the smallest integers such that D is constant in the range [`0(D), n − `1(D)]. This was prov...