$k$-regular mappings of $2^{n}$-dimensional Euclidean space

Abstract:

A map $f:X \to {R^n}$ is said to be k-regular if whenever ${x_1}, \ldots ,{x_k}$ are distinct points of X, then $f({x_1}), \ldots ,f({x_k})$ are linearly independent. Using configuration spaces and homological methods, it is shown that there does not exist a k-regular map from ${R^n}$ into ${R^{n(k - \alpha (k)) + \alpha (k) - 1}}$ where $\alpha (k)$ denotes the number of ones in the dyadic expansion of k and n is a power of 2.