2025/04/08 by Iqra Altaf, Altaf, Iqra, Marianna Csörnyei +1
Mathematics · #28A75 #30L05 #Advanced Banach Space Theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Fixed Point Theorems Analysis #Functional Equations Stability Results
paper · pdf · doi:10.48550/arxiv.2504.06488
openalex publication_date 2025/04/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A generalization of the classical Sard theorem in the plane is the following. Let f be a function defined on a subset A⊂\mathbb R2. If f has modulus of continuity ω(r)\lesssim r2, then f(A)⊂\mathbb R has Lebesgue measure zero. Choquet claimed in \citeChoquet that this was a full characterization, i.e. for every ω for which ω(r)/r2 converges to ∞ as r→ 0, there is a counterexample. We disprove this by showing that the correct characterization, in ℝd, is ∫01 ω(r)-1/d=∞. 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 Rd without decreasing any of the distances in X. That is, we ask the question whether there is an f: X→ \mathbb Rd such that ‖f(x)-f(y)‖≥ ρ(x,y) for every x,y∈ 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 ℝ, and defining ρ by putting ρ(x,y)=r if ‖x-y‖=ω(r).