On the modulus of continuity of functions whose image has positive measure, and metric embeddings into $\mathbb{R}^d$ without shrinking
Iqra Altaf, Marianna Csörnyei
TL;DR
The paper gives a sharp Sard-type characterization for functions whose image has positive measure in ${\mathbb R}^d$ via a modulus of continuity ${\omega}$. It reframes the problem as a metric-embedding question: can a distance-like function ${\rho}$ be realized by a non-shrinking embedding into ${\mathbb R}^d$? It introduces a bounded-growth framework and constructs dyadic-like index sets to build a geometric hierarchy of sets ${C_I},{E_{I,j}},{D_{I,j}}$, linking their interaction to a critical summability condition ${\sum_n 2^{n/d} r_n < \infty}$, which in turn connects to the integral criterion ${\int_0^1} {\omega(r)}^{-1/d}\,dr$. The results yield a sharp threshold: the embedding problem and the Sard-type zero-measure property are governed by this summability/integral condition, with explicit constructions demonstrating sharpness in both Sard-like and Cantor-like scenarios. This provides a precise, dimension-dependent boundary between universally vanishing and potentially positive-measure images under modulus-of-continuity constraints, and it casts the classical Sard phenomenon in a broader metric-embedding framework with quantitative criteria.
Abstract
A generalization of the classical Sard theorem in the plane is the following. Let $f$ be a function defined on a subset $A\subset{\mathbb R}^2$. If $f$ has modulus of continuity $ω(r)\lesssim r^2$, then $f(A)\subset{\mathbb R}$ has Lebesgue measure zero. Choquet claimed in \cite{Choquet} that this was a full characterization, i.e. for every $ω$ for which $ω(r)/r^2$ converges to $\infty$ as $r\to 0$, there is a counterexample. We disprove this by showing that the correct characterization, in $\mathbb{R}^d$, is $\int_{0}^{1} ω(r)^{-1/d}=\infty$. For the precise statement see Theorem 2. We obtain this as a special case of a more general result. We study which spaces $(X,ρ)$ can be embedded into ${\mathbb R}^d$ without decreasing any of the distances in $X$. That is, we ask the question whether there is an $f: X\to {\mathbb R}^d$ such that $\|f(x)-f(y)\|\ge ρ(x,y)$ for every $x,y\in X$. We study this problem for some very general distance functions $ρ$ (we do not even assume that it is a metric space, in particular, we do not assume that $ρ$ satisfies the triangle inequality), and find quantitative necessary and sufficient conditions under which such a mapping exists. We will obtain the characterization mentioned above as a special case of our metric embedding results, by choosing $X$ to be an interval in $\mathbb{R}$, and defining $ρ$ by putting $ρ(x,y)=r$ if $\|x-y\|=ω(r)$.