Around the Danzer Problem and the Construction of Dense Forests

TL;DR

This paper surveys the progress on the Danzer problem and dense forests, discussing various mathematical approaches, generalizing known results, and proposing open problems to advance understanding of these longstanding geometric questions.

Contribution

It provides a comprehensive overview of existing results, extends some of these findings, and formulates open problems to guide future research in the area.

Findings

Survey of diverse mathematical techniques used in the field

Generalization of some known results on dense forests

Identification of open problems for future research

Abstract

A 1965 problem due to Danzer asks whether there exists a set with finite density in Euclidean space intersecting any convex body of volume one. A suitable weakening of the volume constraint leads to the (much more recent) problem of constructing \emph{dense forests}. These are discrete point sets getting uniformly close to long enough line segments. Progress towards these problems have so far involved a wide range of ideas surrounding areas as varied as combinatorial and computation geometry, convex geometry, Diophantine approximation, discrepancy theory, the theory of dynamical systems, the theory of exponential sums, Fourier analysis, homogeneous dynamics, the mathematical theory of quasicrystals and probability theory. The goal of this paper is to survey the known results related to the Danzer Problem and to the construction of dense forests, to generalise some of them and to state a number of open problems to make further progress towards a solution to this longstanding question.