We show that, for every set of n points in the d-dimensional unit cube, there is an empty axis-parallel box of volume at least Ω(d/n) as n → ∞ and d is fixed. In the opposite direction, we give a construction without an empty axis-parallel box of volume O(d 2 log d/n). These improve on the previous best bounds of Ω(log d/n) and O(2 7d /n) respectively.

Given a finite set $A \subseteq \mathbb{R}^d$ , points $a_1,a_2,\dotsc,a_{\ell} \in A$ form an $\ell$ -hole in A if they are the vertices of a convex polytope, which contains no points of A in its interior. We construct arbitrarily large point sets in general position in $\mathbb{R}^d$ having no holes of size $O(4^dd\log d)$ or more. This improves the previously known upper bound of order $d^{d+o(d)}$ due to Valtr. The basic version of our construction uses a certain type of equidistributed point sets, originating from numerical analysis, known as (t,m,s)-nets or (t,s)-sequences, yielding a bound of $2^{7d}$ . The better bound is obtained using a variant of (t,m,s)-nets, obeying a relaxed equidistribution condition.

Let ${\bf s} = (s_1, s_2, \ldots, s_n,\ldots)$ be a sequence of positive integers. An ${\bf s}$-inversion sequence of length $n$ is a sequence ${\bf e} = (e_1, e_2, \ldots, e_n)$ of nonnegative integers such that $0 \leq e_i < s_i$ for $1\leq i\leq n$. When $s_i=(i-1)k+1$ for any $i\geq 1$, we call the ${\bf s}$-inversion sequences the $k$-inversion sequences. In this paper, we provide a bijective proof that the ascent number over $k$-inversion sequences of length $n$ is equidistributed with a weighted variant of the ascent number of permutations of order $n$, which leads to an affirmative answer of a question of Savage (2016). A key ingredient of the proof is a bijection between $k$-inversion sequences of length $n$ and $2\times n$ arrays with particular restrictions. Moreover, we present a bijective proof of the fact that the ascent plateau number over $k$-Stirling permutations of order $n$ is equidistributed with the ascent number over $k$-inversion sequences of length $n$.

Given a finite set A ⊆ R d , points a 1 , a 2 , . . . , a ℓ ∈ A form an ℓ-hole in A if they are the vertices of a convex polytope which contains no points of A in its interior. We construct arbitrarily large point sets in general position in R d having no holes of size 2 7d or more. This improves the previously known upper bound of order d d+o(d) due to Valtr. Our construction uses a certain type of equidistributed point sets, originating from numerical analysis, known as (t, m, s)-nets or (t, s)-sequences. IntroductionPoints a 1 , a 2 , . . . , a ℓ ∈ A are in convex position if they are the vertices of a convex polytope. If that polytope is empty, i.e., contains no points of A in its interior, the points a 1 , a 2 , . . . , a ℓ are said to form an ℓ-hole in A.A classic result of Erdős and Szekeres [ES35] asserts that for any positive integer ℓ, every sufficiently large finite set A in general position in R 2 contains ℓ points in convex position. Erdős [Erd75] went on to ask if one can also guarantee an ℓ-hole in a large enough A ⊆ R 2 in general position. Harborth [Har78] proved that one can always find a 5-hole, while Horton [Hor83] constructed arbitrarily large sets without any 7-hole. The remaining case ℓ = 6 turned out to be more challenging, but was settled in the affirmative by Nicolás [Nic07] and, independently, Gerken [Ger08].Another question studied is the asymptotic behavior, as n → ∞, of the number of ℓ-holes guaranteed to exist in a set A of n points in general position in R 2 . For ℓ = 3, 4 this number was shown to be Θ(n 2 ) by Katchalski and Meir [KM88] and Bárány and Füredi [BF87]. The order of magnitude for ℓ = 5, 6 is not known, but very recently Aichholzer et al. [ABH + 20] proved it is superlinear for ℓ = 5. Turning to higher dimensions, much less is known. Valtr [Val92] gave a simple projection argument to extend the Erdős-Szekeres result to any dimension d ≥ 2: for every ℓ, any sufficiently large finite set A in general position in R d contains ℓ points in convex position. Regarding holes, he defined: h(d) def = max{ℓ : any large enough A ⊆ R d in general position contains an ℓ-hole}.

We show that, for every set of $n$ points in the $d$-dimensional unit cube, there is an empty axis-parallel box of volume at least $\Omega (d/n)$ as $n\to \infty $ and $d$ is fixed. In the opposite direction, we give a construction without an empty axis-parallel box of volume $O(d^2\log d/n)$. These improve on the previous best bounds of $\Omega (\log d/n)$ and $O(2^{7d}/n)$, respectively.