On the dispersion of sparse grids

For any natural number d and positive number ε, we present a point set in the d-dimensional unit cube [0, 1] that intersects every axis-aligned box of volume greater than ε. These point sets are very easy to handle and in a vast range for ε and d, we do not know any smaller set with this property. 1 The Result The dispersion of a point set P in [0, 1] is the volume of the largest axis-aligned box in [0, 1] which does not intersect P . Point sets with small dispersion already proved to be useful for the uniform recovery of rank one tensors [4] and for the discretization of the uniform norm of trigonometric polynomials [9]. Recently, great progress has been made in the question for the minimal size for which there exists a point set whose dispersion is at most ε > 0, see Dumitrescu and Jiang [3], Aistleitner, Hinrichs and Rudolf [1], Rudolf [7] and Sosnovec [8]. In this note, we want to provide such point sets. They should be both small and easy to handle. To that end, we define the point sets P (k, d) = ⋃ |j|=k Mj1 × · · · ×Mjd of order k ∈ N0 and dimension d ∈ N, generated by the one-dimensional sets Mj = { 1 2j+1 , 3 2j+1 , . . . , 2 − 1 2j+1 } for j ∈ N0. You can find a picture of the set of order 3 in dimension 2 in Figure 1. These point sets are particular instances of a sparse grid as widely used for high-dimensional numerical integration and approximation. We refer to Novak and Woźniakowski [5] and the references therein. Here, we will prove the following result. 1 Theorem. Let k(ε) = ⌈log2 (ε )⌉ − 1 for any ε ∈ (0, 1) and let d ≥ 2. Then the dispersion of P (k(ε), d) is at most ε and |P (k(ε), d)| = 2 (