arXiv subjects

Jaspar Wiart

Publications and source records attributed to Jaspar Wiart.

5 recordsLinked to original sources

arXiv2023

Golden Ratio Nets and Sequences

In this paper we introduce and study nets and sequences constructed in an irrational base, focusing on the case of a base given by the golden ratio $ϕ$. We provide a complete framework to study equidistribution properties of nets in base $ϕ$, which among other things requires the introduction of a new concept of prime elementary intervals which differ from the standard definition used for integer bases. We define the one-dimensional van der Corput sequence in base $ϕ$ and two-dimensional Hammersley point sets in base $ϕ$ and we prove some properties for $(0,1)-$sequences and $(0,m,2)-$nets in base $ϕ$ respectively. We also include numerical studies of the discrepancy of point sets and sequences in base $ϕ$ showing an improvement in distribution properties over traditional integer based Hammersley constructions. As motivation for future research, we show how the equidistribution notions that are introduced for base $ϕ$ can be generalized to other irrational bases.

math.NT↗

arXiv2021

On the area of empty axis-parallel rectangles amidst 2-dimensional lattice points

The dispersion of a point set in the unit square is defined to be the area of the largest empty axis-parallel box. In this paper we are interested in the dispersion of lattices in the plane, that is, the supremum of the area of the empty axis-parallel boxes amidst the lattice points. We introduce a framework with which to study this based on the continued fractions expansions of the generators of the lattice. This framework proves so successful that we were unable to ask a question that we could not answer. We give necessary and sufficient conditions under which a lattice has finite dispersion. We obtain an exact formula for the dispersion of the lattices associated to subgroups of the ring of integer of a quadratic field. We have tight bounds for the dispersion of a lattice based the largest continued fraction coefficient of the generators, accurate to within one half. We know what the $n$-th best lattice is. We provide an equivalent formulation of Zaremba's conjecture. Using our framework we are able to give alternative proofs of the results from two other papers in only a few lines.

math.NT↗

arXiv2020

Improved dispersion bounds for modified Fibonacci lattices

We study the dispersion of point sets in the unit square; i.e. the size of the largest axes-parallel box amidst such point sets. It is known that $\liminf_{N\to\infty} N\mathrm{disp}(N,2)\in \left[\frac54,2\right],$ where $\mathrm{disp}(N,2)$ is the minimal possible dispersion for an $N$-element point set in the unit square. The upper bound 2 is obtained by an explicit point construction - the well-known Fibonacci lattice. In this paper we find a modification of this point set such that its dispersion is significantly lower than the dispersion of the Fibonacci lattice. Our main result will imply that $\liminf_{N\to\infty} N\mathrm{disp}(N,2)\leq φ^3/\sqrt{5}=1.894427...$

math.CO↗

arXiv2020

On the dependence structure and quality of scrambled $(t,m,s)-$nets

In this paper we develop a framework to study the dependence structure of scrambled $(t,m,s)$-nets. It relies on values denoted by $C_b(\mathbf{k};P_n)$, which are related to how many distinct pairs of points from $P_n$ lie in the same elementary $\mathbf{k}-$interval in base $b$. These values quantify the equidistribution properties of $P_n$ in a more informative way than the parameter $t$. They also play a key role in determining if a scrambled set $\tilde{P}_n$ is negative lower orthant dependent (NLOD). Indeed this property holds if and only if $C_b(\mathbf{k};P_n) \le 1$ for all $\mathbf{k} \in \mathbb{N}^s$, which in turn implies that a scrambled digital $(t,m,s)-$net in base $b$ is NLOD if and only if $t=0$. Through numerical examples we demonstrate that these $C_b(\mathbf{k};P_n)$ values are a powerful tool to compare the quality of different $(t,m,s)$-nets, and to enhance our understanding of how scrambling can improve the quality of deterministic point sets.

math.PR↗

arXiv2020

Walsh functions, scrambled $(0,m,s)$-nets, and negative covariance: applying symbolic computation to quasi-Monte Carlo integration

We investigate base $b$ Walsh functions for which the variance of the integral estimator based on a scrambled $(0,m,s)$-net in base $b$ is less than or equal to that of the Monte-Carlo estimator based on the same number of points. First we compute the Walsh decomposition for the joint probability density function of two distinct points randomly chosen from a scrambled $(t,m,s)$-net in base $b$ in terms of certain counting numbers and simplify it in the special case $t$ is zero. Using this, we obtain an expression for the covariance of the integral estimator in terms of the Walsh coefficients of the function. Finally, we prove that the covariance of the integral estimator is negative when the Walsh coefficients of the function satisfy a certain decay condition. To do this, we use creative telescoping and recurrence solving algorithms from symbolic computation to find a sign equivalent closed form expression for the covariance term.

math.NA↗