Roberts-type embeddings and conversion of transversal Tverberg's theorem

Central in the paper are two results on the existence of “economical” embeddings in a Euclidean space. The first result (Corollary 1.4) states the existence of an embedding with image intersecting the large-dimensional planes in sets of “controllable” dimension. The second result (Corollary 1.6) proves the existence of maps such that each small-dimensional plane contains “controllably” many points of the image.
Well known results of Nöbeling–Pontryagin, Roberts, Hurewicz, Boltyanskii, and Goodsell can be obtained as consequences of these results. Their infinite-dimensional version concerning an embedding in a Hilbert space is also established (Theorem 1.8).