A new simple family of Cantor sets in $\mathbb{R}^3$ all of whose projections are one-dimensional

Abstract

In 1994, J.Cobb described a Cantor set in $\mathbb{R}^3$ each of whose projections into 2-planes is one-dimensional. A series of works by other authors developing this field followed. We present another very simple series of Cantor sets in $\mathbb{R}^3$ all of whose projections are connected and one-dimensional. These are self-similar Cantor sets which go back to the work of Louis Antoine, and we celebrate their centenary birthday in 2020-2021.