Discrepancy of the sequence \(\Big(\Big\{n{1+\sqrt{5}\over 2}\Big\}\Big)\). (Discrépance de la suite \(\Big(\Big\{n{1+\sqrt{5}\over 2}\Big\}\Big)\).) (French) Zbl 0386.10021

Let \(D^*(N)\) be the star-discrepancy of the sequence \(\Big(\Big\{n{1+\sqrt{5}\over 2}\Big\}\Big)\). We show that
\[ \limsup {D^*(N)\over \log N} = {3\over 20} \Big(\log{1+\sqrt 5\over 2}\Big) ^{-1} = 0.31\cdots, \] which illustrates the fact that our sequence has smaller star-discrepancy than that of van der Corput’s sequence. Our proofs involve continued fraction theory.
Reviewer: Yves Dupain


11K06 General theory of distribution modulo \(1\)
11K31 Special sequences
Full Text: DOI Numdam EuDML


[1] [1] et , Discrépance de la suite de van der Corput, C.R. Acad. Sc., Paris, 285 (1977), 313-316. · Zbl 0361.10032
[2] [2] , Intervalles à restes majorés pour la suite {nα}, Acta. Math. Acad. Scient. Hung., t. 29 (3,4) (1977), 289-303. · Zbl 0372.10026
[3] [3] and , Uniform distribution of sequences, Wiley Interscience, New York, (1974), 88-132. · Zbl 0281.10001
[4] [4] , Sur la répartition modulo 1 des suites {nα}, Acta Arith., 20 (1972), 345-352. · Zbl 0239.10018
[5] [5] , Sur la répartition modulo 1 des suites {nα}, Séminaire Delange-Pisot-Poitou, (1966-1967), fascicule 1, exposé n° 2. · Zbl 0164.05502
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.