How many sprays cover the space?

TL;DR

This paper extends previous results on covering high-dimensional spaces with sprays, establishing a relationship between the cardinality of real numbers and the number of sprays needed in higher dimensions.

Contribution

It generalizes Schmerl's result from 2D to all dimensions d ≥ 3, linking space coverage with cardinality constraints.

Findings

For all d ≥ 3, the space can be covered with (n+1)(d-1)+1 sprays in certain conditions.

The cardinality of the real numbers is at most ℵ_n under specific space covering conditions.

Extension of previous 2D results to higher dimensions.

Abstract

For all $d \geq 3$ we show that the cardinality of $ \mathbb{R} $ is at most $\aleph_n $ if and only if $ \mathbb{R}^d $ can be covered with $ ( n + 1 ) ( d - 1 ) + 1 $ sprays whose centers are in general position in a hyperplane. This extends previous results by Schmerl when $ d = 2 $.