Regularities in the distribution of special sequences

2020

Let (H(n))n≥0 be a 2−dimensional Halton’s sequence. Let D2((H(n)) N−1 n=0 ) be the L2-discrepancy of (Hn) N−1 n=0 . It is known that lim supN→∞(logN) D2(H(n)) N−1 n=0 > 0. In this paper, we prove that D2((H(n)) N−1 n=0 ) = O(logN) for N →∞, i.e., we found the smallest possible order of magnitude of L2-discrepancy of a 2-dimensional Halton’s sequence. The main tool is the theorem on linear forms in the p-adic logarithm.

Comptes Rendus Mathematique, 2016

Let (H s (n)) n≥1 be an s-dimensional Halton's sequence. Let D N be the discrepancy of the sequence (H s (n)) N n=1. It is known that N D N = O (ln s N) as N → ∞. In this paper, we prove that this estimate is exact: lim N→∞ N ln −s (N)D N > 0.

Ergodic Theory and Dynamical Systems, 2013

A ($d$-parameter) basic nilsequence is a sequence of the form $\psi (n)= f({a}^{n} x)$,$n\in { \mathbb{Z} }^{d} $, where $x$ is a point of a compact nilmanifold $X$, $a$ is a translation on $X$, and$f\in C(X)$; a nilsequence is a uniform limit of basic nilsequences. If $X= G/ \Gamma $ is a compact nilmanifold, $Y$ is a subnilmanifold of $X$, $\mathop{(g(n))}\nolimits_{n\in { \mathbb{Z} }^{d} } $ is a polynomial sequence in $G$, and $f\in C(X)$, we show that the sequence $\phi (n)= \int \nolimits \nolimits_{g(n)Y} f$ is the sum of a basic nilsequence and a sequence that converges to zero in uniform density (a null-sequence). We also show that an integral of a family of nilsequences is a nilsequence plus a null-sequence. We deduce that for any invertible finite measure preserving system $(W, \mathcal{B} , \mu , T)$, polynomials ${p}_{1} , \ldots , {p}_{k} : { \mathbb{Z} }^{d} \longrightarrow \mathbb{Z} $, and sets ${A}_{1} , \ldots , {A}_{k} \in \mathcal{B} $, the sequence $\phi (n)= ...

arXiv (Cornell University), 2019

Let x 0 , x 1 , ... be a sequence of points in [0, 1) s. A subset S of [0, 1) s is called a bounded remainder set if there exist two real numbers a and C such that, for every integer N , |card{n < N | x n ∈ S} − aN | < C. Let (x n) n≥0 be an s−dimensional Halton-type sequence obtained from a global function field, b ≥ 2, γ = (γ 1 , ..., γ s), γ i ∈ [0, 1), with b-adic expansion γ i = γ i,1 b −1 + γ i,2 b −2 + ..., i = 1, ..., s. In this paper, we prove that [0, γ 1) × ... × [0, γ s) is the bounded remainder set with respect to the sequence (x n) n≥0 if and only if max 1≤i≤s max{j ≥ 1 | γ i,j = 0} < ∞. We also obtain the similar results for a generalized Niederreiter sequences, Xing-Niederreiter sequences and Niederreiter-Xing sequences.

Mathematics and Computers in Simulation, 2005

Quasi-Monte Carlo methods are a variant of ordinary Monte Carlo methods that employ highly uniform quasirandom numbers in place of Monte Carlo’s pseudorandom numbers. Clearly, the generation of appropriate high-quality quasirandom sequences is crucial to the success of quasi-Monte Carlo methods. The Halton sequence is one of the standard (along with (t,s)(t,s)-sequences and lattice points) low-discrepancy sequences, and one of its important advantages is that the Halton sequence is easy to implement due to its definition via the radical inverse function. However, the original Halton sequence suffers from correlations between radical inverse functions with different bases used for different dimensions. These correlations result in poorly distributed two-dimensional projections. A standard solution to this phenomenon is to use a randomized (scrambled) version of the Halton sequence. An alternative approach to this is to find an optimal Halton sequence within a family of scrambled sequences. This paper presents a new algorithm for finding an optimal Halton sequence within a linear scrambling space. This optimal sequence is numerically tested and shown empirically to be far superior to the original. In addition, based on analysis and insight into the correlations between dimensions of the Halton sequence, we illustrate why our algorithm is efficient for breaking these correlations. An overview of various algorithms for constructing various optimal Halton sequences is also given.

Mathematica Slovaca, 2004

Properties of distribution functions of block sequences were investigated in [STRAUCH, O .—TOTH, J. T.: Distribution functions of ratio sequences, Publ . Math. Debrecen 58 (2001), 751-778]. The present paper is a continuation of the study of relations between the density of the block sequence and so called dispersion of the block sequence. Preliminaries In this part we recall some basic definitions. Denote by N and M the set of all positive integers and positive real numbers, respectively. For X C N let X(n) = card{x G X : x &lt; n}. In the whole paper we will assume that X is infinite. Denote by R(X) = { | : x G X, y G X} the ratio set of X and say that a set X is (R) -dense if R(X) is (topologically) dense in the set M . Let us notice that the concept of (It)-density was defined and first studied in papers [SI] and [S2]. Now let X = {x1,x2,...} where xn &lt; x n + 1 are positive integers. The following sequence of finite sequences derived from X X rp rp rp rp rp rp rp rv* (1) ^1 ^...

Discrete Mathematics, 2005

A sequence (a i) of integers is weak Sidon or well-spread if the sums a i + a j , for i < j, are all different. Let f (N) denote the maximum integer n for which there exists a weak Sidon sequence 0 a 1 < • • • < a n N. Using an idea of Lindström [An inequality for B 2-sequences, J. Combin. Theory 6 (1969) 211-212], we offer an alternate proof that f (N) < N 1/2 + O(N 1/4), an inequality due to Ruzsa [Solving a linear equation in a set of integers I, Acta. Arith. 65 (1993) 259-283]. The present proof improves Ruzsa's bound by decreasing the implicit constant, essentially from 4 to √ 3.

This paper has several folds. We make first new permu- tation choices in Halton sequences to improve their distributions. These choices are multi-dimensional and they are made for two different discrep- ancies. We show that multi-dimensional choices are better for standard quasi-Monte Carlo methods. We also use these sequences as a variance re- duction technique in Monte Carlo methods, which greatly improves the convergence accuracy of the estimators. For this kind of use, we observe that one-dimensional choices are more efficient.

Periodica Mathematica Hungarica, 2016

We consider variants of the Halton sequence in a generalized numeration system, called the Cantor expansion. We show that it provides a wealth of low-discrepancy sequences by giving an estimate of the (star) discrepancy of the Halton sequence in each bounded Cantor base. The techniques used in our estimation of the discrepancy are adapted from those developed by E.I. Atanassov.

2015

Let ((n))_n ≥ 1 be an s-dimensional Niederreiter-Xing sequence in base b. Let D(((n))_n = 1^N) be the discrepancy of the sequence ((n))_n = 1^N . It is known that N D(((n))_n = 1^N) =O(^s N) as N →∞ . In this paper, we prove that this estimate is exact. Namely, there exists a constant K&gt;0, such that _∈ [0,1)^s_1 ≤ N ≤ b^m N D(((n)⊕)_n = 1^N) ≥ K m^s for m=1,2,... . We also get similar results for other explicit constructions of (t,s) sequences.