Low Discrepancy Digital Kronecker-Van Der Corput Sequences

The discrepancy of a sequence measures how quickly it approaches a uniform distribution. Given a natural number d, any collection of one dimen- sional so-called low discrepancy sequences {S i : 1 ≤ i ≤ d} can be concate- nated to create a d-dimensional hybrid sequence (S 1 , . . . , S d). Since their intro- duction by Spanier in 1995, many connections between the discrepancy of a hybrid sequence and the discrepancy of its component sequences have been discovered. However, a proof that a hybrid sequence is capable of being low discrepancy has remained elusive. The purpose of this note is to remedy this by providing an ex- plicit connection between Diophantine approximation over function fields and two dimensional low discrepancy hybrid sequences. Specifically, let F q be the finitefield of cardinality q. It is shown that some real numbered hybrid sequence H Θ(t), P (t) := H(Θ, P ) built from the dig- ital Kronecker sequence associated to a Laurent series Θ(t) ∈ F q((t−1)) and the digital Van der Corput sequence associated to an irreducible polynomial P (t) ∈ F q[t] meets the above property. More precisely, if Θ(t) is a counterexample to the so called t-adic Littlewood Conjecture (t-LC), then another Laurent series Φ(t) ∈ F q((t−1)) induced from Θ(t) and P (t) can be constructed so that H(Φ, P ) is low discrepancy. Such counterexamples to t-LC are known over a number of fi- nite fields by, on the one hand, Adiceam, Nesharim and Lunnon, and on the other, by Garrett and the author.