Danzer's problem, effective constructions of dense forests and digital sequences

A 1965 problem due to Ludwig Danzer asks whether there exists a set in Euclidean space with finite density intersecting any convex body of volume 1. recent approach this is concerned the construction dense forests and obtained by suitable weakening constraint. forest discrete point getting uniformly close long enough line segments. The distribution points then quantified terms visibility function. Another way weaken assumptions Danzer's relaxing In respect, new concept introduced paper, namely that an optical forest. An R d $\mathbb {R}^{d}$ optimal but not necessarily density. literature, best constructions sets are deterministic. goal paper provide deterministic which yield known results dimension ⩾ 2 $d \geqslant 2$ bounds, respectively. Namely, three main work: (1) bound which, furthermore, enjoys property being deterministic; (2) failing be only up logarithm (3) planar Peres-type (that is, from Peres) bound. This achieved constructing digital sequence satisfying strong dispersion properties.