On the Steinhaus tiling problem in three dimensions

TL;DR

This paper investigates the existence of Steinhaus sets in three-dimensional space, providing heuristic evidence suggesting such sets do not exist, extending the classical problem from two to three dimensions.

Contribution

It offers the first heuristic analysis indicating the non-existence of Steinhaus sets in three dimensions, building on prior two-dimensional constructions.

Findings

Heuristic evidence against the existence of Steinhaus sets in R^3.

Extension of Steinhaus problem from 2D to 3D.

Supports the conjecture that no such sets exist in three dimensions.

Abstract

H. Steinhaus asked in the 1950's whether there exists a set in the plane R^2 meeting every isometric copy of Z^2 in precisely one point. Such a "Steinhaus set" was constructed by Jackson and Mauldin. What about three-space R^3? Is there a subset of R^3 meeting every isometric copy of Z^3 in exactly one point? We offer heuristic evidence that the answer is "no".