Decomposing with smooth sets

Abstract

A subset of Euclidean space will be said to be $n$-smooth if it has an $n$-dimensional tangent plane at each of its points. Let ${\frak d}_n$ denote the least number $n$-smooth sets into which $n+1$-dimensional Euclidean space can be decomposed. For each $n$ it is shown to be consistent that ${\frak d}_n > {\frak d}_{n+1} $. Moreover, the inequalities ${\frak d}_{n+1}^+ \geq ${\frak d}_n$ are established where ${\frak d}_1$ is defined to be the continuum. The cardinal invariant ${\frak d}_2$ is shown to be the same as the least $\kappa$ such that each continuous function from the reals to the reals can be decomposed into $\kappa$ differentiable functions.