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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
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