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2014

On the lower bound of the discrepancy of Halton's sequence

Levin, Mordechay B.

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Let $ (H_s(n))_{n \geq 1} $ be an $s-$dimensional Halton's sequence.

Let $D_N$ be the discrepancy of the sequence $ (H_s(n))_{n = 1}^{N} $.
It is known that $ND_N =O(\ln^s N)$ as $N \to \infty $.
In this paper we prove that this estimate is exact: $$ \overline{\lim}_{ N \to \infty} N \ln^{-s}(N) D_N >0.

On the lower bound of the discrepancy of Halton's sequence · Around

Built on

Beck, J., Chen, W. W. L., Irregularities of Distribution, Cambridge Univ. Press, Cambridge, 1987
1987
Earlier work this paper cites.
Faure, H., Chaix, H., Minoration de discrépance en dimension deux, Acta Arith. 76 (1996), no. 2, 149-164
1996
Earlier work this paper cites.
Drmota, M., Tichy, R., Sequences, Discrepancies and Applications, Lecture Notes in Mathematics 1651, 1997
1997
Earlier work this paper cites.

Similar

Bilyk, D., On Roth’s orthogonal function method in discrepancy theory. Unif. Distrib. Theory 6 (2011), no. 1, 143-184
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Levin M.B., On the lower bound of the discrepancy of ( t , s ) (t,s) sequence, arxiv.org, January 2015
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Then

Levin M.B., On the lower bound of the discrepancy of Halton’s sequence, II, arxiv.org, January 2015
2015
Closest in time.
Levin M.B., On the lower bound in the lattice point remainder problem for a parallelepiped, http://arxiv.org/abs/1307.2080
Original
2080
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