Cantor sets with high-dimensional projections

Most Cantor sets in $\mathbb R^N$ are in general position with respect to all projections

“…In 1906, L. Zoretti recalls as a known fact ("ce resultat bien connu etant acquis...") that the graph of any such surjection is a Cantor set in plane whose projection to y-axis coincides with I. (See [7] or [8] for historic details. )…”

“…1]; (m, n, n) [3]; (3, 2, 1) [4]; (m, n, n − 1) [6]; (m, m − 1, k) [2]. Other constructions for the cases (m, m − 1, m − 2) and (3, 2, 1) are described in the papers [7] and [9] which also contain further references. For each integer n, there is a Cantor set in the Hilbert space ℓ 2 all of whose projections into n-planes are (n − 1)-dimensional [2].…”

Most Cantor sets in $\mathbb R^N$ are in general position with respect to all projections

“…1]; (m, n, n) [3]; (3, 2, 1) [4]; (m, n, n − 1) [6]; (m, m − 1, k) [2]. Other constructions for the cases (m, m − 1, m − 2) and (3, 2, 1) are described in [7] and [9], which also contain further references. For each integer n, there is a Cantor set in the Hilbert space ℓ 2 all of whose projections into n-planes are (n − 1)-dimensional [2].…”