Autor(es)

Verónica Becher and Olivier Carton

URL Pública

Abstract

MB Levin used Sobol–Faure low discrepancy sequences with Pascal triangle matrices modulo 2 to construct, a real number x such that the first N terms of the sequence (2 n x mod 1) n≥ 1 have discrepancy O ((log N) 2∕ N). This is the lowest discrepancy known for this kind of sequences. In this note we characterize Levin’s construction in terms of nested perfect necklaces, which are a variant of the classical de Bruijn sequences. Moreover, we show that every real number x whose binary expansion is the concatenation of nested perfect necklaces of exponentially increasing order satisfies that the first N terms of (2 n x mod 1) n≥ 1 have discrepancy O ((log N) 2∕ N). For the order being a power of 2, we give the exact number of nested perfect necklaces and an explicit method based on matrices to construct each of them. The computation of the n th digit of the binary expansion of a real number built from …