On Cantor sets with shadows of prescribed dimension

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

“…Concerning the case of ℓ 2 , it is appropriate to remind the reader of Theorem 3 from [2] which says that for any integer k and any compact subset M ⊂ ℓ 2 , the restriction p H | M is an embedding for all H in some dense subset of the Grassmann space G k (ℓ 2 ).…”

“…2, p.273], [4, p. 124, Example], [5,Prop. 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.…”

“…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

“…273], [4, p. 124, Example], [5,Prop. 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.…”

“…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].…”