On the sharpness of assumptions in the Federer theorem

Abstract:

The Federer theorem deals with the “massiveness” of the set of critical values for a $t$-smooth map acting from $\mathbb R^m$ to $\mathbb R^n$: it claims that the Hausdorff $p$-measure of this set is zero for certain $p$. If $n\ge m$, it has long been known that the assumption of that theorem relating the parameters $m,n,t,p$ is sharp. Here it is shown by an example that this assumption is also sharp for $n<m$.