Cyclic shifts of the van der Corput set

TL;DR

This paper constructs a specific cyclic shift of the van der Corput set that achieves minimal $L^2$ discrepancy, building on prior theoretical bounds and providing an explicit example.

Contribution

It provides an explicit construction of a cyclic shift of the van der Corput set with minimal $L^2$ discrepancy, complementing previous existence results.

Findings

Constructed a specific cyclic shift with minimal $L^2$ discrepancy

Confirmed the theoretical bounds on discrepancy are attainable with explicit examples

Enhances understanding of discrepancy properties of shifted van der Corput sets

Abstract

In [13], K. Roth showed that the expected value of the $L^2$ discrepancy of the cyclic shifts of the $N$ point van der Corput set is bounded by a constant multiple of $\sqrt{\log N}$, thus guaranteeing the existence of a shift with asymptotically minimal $L^2$ discrepancy, [11]. In the present paper, we construct a specific example of such a shift.