Embeddings of finite-dimensional compacta in Euclidean spaces

If g is a map from a space X into R m and q is an integer, let B q , d , m ( g ) be the set of all planes Π d ⊂ R m such that | g − 1 ( Π d ) | ⩾ q . Let also H ( q , d , m , k ) denote the maps g : X → R m such that dim B q , d , m ( g ) ⩽ k . We prove that for any n-dimensional metric compactum X each of the sets H ( 3 , 1 , m , 3 n + 1 − m ) and H ( 2 , 1 , m , 2 n ) is dense and G δ in the function space C ( X , R m ) provided m ⩾ 2 n + 1 (in this case H ( 3 , 1 , m , 3 n + 1 − m ) and H ( 2 , 1 , m , 2 n ) can consist of embeddings). The same is true for the sets H ( 1 , d , m , n + d ( m − d ) ) ⊂ C ( X , R m ) if m ⩾ n + d , and H ( 4 , 1 , 3 , 0 ) ⊂ C ( X , R 3 ) if dim X ⩽ 1 . This result complements an authorsʼ result from Bogatyi and Valov (2005) [5]. A parametric version of the above theorem, as well as a partial answer of a question from Bogatyi (2008) [4] and Bogatyi and Valov (2005) [5] are also provided.