Low-discrepancy sequences: Atanassov's methods revisited

We revisit Atanassov's methods for Halton sequences, (t,s)-sequences, and (t,e,s)-sequences by providing a unifying framework enlightening the power and the efficiency of these methods for the study of low-discrepancy sequences. In this context, we obtain new improved explicit bounds for the star-discrepancy of these sequences, showing in most cases a better behavior than preceding ones in the non-asymptotic regime. Theoretical comparisons of discrepancy bounds in the non-asymptotic regime are much more difficult to achieve than in the asymptotic regime, where results exist to compare the leading constants cs. Hence in this paper we mostly proceed via numerical comparisons to compare bounds. But in the case of (t,s)-sequences in base 2, we are able to compare two discrepancy bounds and prove that one is demonstrably better than the other for any N≥2s. The proof is far from trivial as the two bounds are based on different combinatorial arguments.