Netyjnd

Let $f: mathbbR^mrightarrow mathbbR^m-k$ be a Lipschitz map.

Can we get a uniform estimate on the Hausdorff dimension of the boundaries of fibres of $f$? I.e. do we have an upper bound for
$$ sup_yin mathbbR^n-k dim_H(partial f^-1(y)) ?$$

Theorem 2.5 in [1] tells us, that for almost every $yin mathbbR^n-k$ we have that $dim_H(f^-1(y))leq k$. This tells us
$$ textessup_yin mathbbR^n-k dim_H(partial f^-1(y)) leq k.$$
Can we pass to the supremum? And are there even better bounds? I mean, I used $partial f^-1(y)subseteq f^-1(y)$ as $f$ is continuous and the monotonicity of the Hausdorff dimension, but I guess that one can do better than this.

[1] G. Alberti, S. Bianchini, G. Crippa, Structure of level sets and Sard-type properties of Lipschitz maps: results and counterexamples.
Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 12 (2013), no. 4, 863–902.

Unfortunately, you can always find a Lipschitz map
$$
f:mathbbR^mtomathbbR^m-k
quad
textand
quad
yinmathbbR^m-k
$$
such that $partial f^-1(y)$ has positive $m$-dimensional measure so
$dim_H partial f^-1(y)=m$.

Here is an example. Let $KsubsetmathbbR^m$ be a Cantor set (i.e. a set homeomorphic to the ternary Cantor set) of positive $m$-dimensional measure. Existence of such a set $K$ is standard. Let $f(x)=operatornamedist(x,K)$. Then $f:mathbbR^mtomathbbR$ is $1$-Lipschitz and it vanishes precisely on $K$. That is $f^-1(0)=K=partial K$ (the boundary of a Cantor set is the Cantor set itself) has positive $m$-dimensional measure. Now, assuming that $mathbbRsubsetmathbbR^m-k$ we can regard $f$ as a mapping $f:mathbbR^mtomathbbR^m-k$.