## 25 Citations

### Empty axis-parallel boxes

- MathematicsArXiv
- 2020

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…

### An Upper Bound of the Minimal Dispersion via Delta Covers

- MathematicsArXiv
- 2017

It is proved that there is a point set, such that the largest volume of such a test set without any point is bounded above by \(\frac {\log \vert \varGamma _\delta \vert }{n} + \delta}\).

### N A ] 3 0 O ct 2 01 7 Fixed volume discrepancy in the periodic case

- Mathematics
- 2018

Discrepancy theory is a classical well established area of research in geometry and numerical integration (see [2], [8], [15], [17]). Recently, in [18], a new phenomenon has been discovered. A…

### Piercing All Translates of a Set of Axis-Parallel Rectangles

- MathematicsIWOCA
- 2021

This work requires that a single point set P simultaneously pierces each translate of each shape from some family F, and denotes the lowest possible density of such an F-piercing point set by πT (F).

## References

SHOWING 1-10 OF 16 REFERENCES

### An Upper Bound of the Minimal Dispersion via Delta Covers

- MathematicsArXiv
- 2017

It is proved that there is a point set, such that the largest volume of such a test set without any point is bounded above by \(\frac {\log \vert \varGamma _\delta \vert }{n} + \delta}\).

### On the Largest Empty Axis-Parallel Box Amidst n Points

- Computer Science, MathematicsAlgorithmica
- 2012

We give the first efficient (1−ε)-approximation algorithm for the following problem: Given an axis-parallel d-dimensional box R in ℝd containing n points, compute a maximum-volume empty…

### Computing the Largest Empty Rectangle

- Computer ScienceSTACS
- 1984

A divide-and-conquer approach similar to the ones used by Strong and Bentley is used and a new notion of Voronoi diagram is introduced along with a method for efficient computation of certain functions over paths of a tree.

### Tractability of the Approximation of High-Dimensional Rank One Tensors

- Computer Science, Mathematics
- 2014

It is proved that for certain parameters (smoothness and norm of the $$r$$rth derivative), this problem is intractable, while for other parameters, the problem is tractable and the complexity is only polynomial in the dimension for every fixed $$\varepsilon >0$$ε>0.

### Number Theory, Fourier Analysis and Geometric Discrepancy

- Mathematics
- 2014

Part I. Elementary Number Theory: 1. Prelude 2. Arithmetic functions and integer points 3. Congruences 4. Quadratic reciprocity and Fourier series 5. Sums of squares Part II. Fourier Analysis and…

### The Marcinkiewicz-Type Discretization Theorems

- Mathematics, Computer ScienceConstructive Approximation
- 2018

A new technique is presented, which works well for discretization of the integral norm, which is a combination of probabilistic technique, based on chaining, and results on the entropy numbers in the uniform norm.

### Approximating Integrals Via Monte Carlo and Deterministic Methods

- Mathematics
- 2000

This book is designed to introduce graduate students and researchers to the primary methods useful for approximating integrals, and although the focus is on higher- dimensional integrals the lower-dimensional case is also covered.