A rank zero theorem related to the Hausdorff measure (Q2719662)

The main result of this paper is to prove a rank zero theorem concerned with Hausdorff measures.NEWLINENEWLINENEWLINETheorem. Let \(k,m,n\) be positive integers, \(\alpha\in [0,1]\), and \(d= m/(k+ \alpha)\), \(k+\alpha\neq 1\). If \(f: \mathbb{R}^m\to \mathbb{R}^n\) is a \(C^{k,\alpha}\)-function and \(A= C_0(f)\), then \(f(A)\) is of \(d\)-zero measure.NEWLINENEWLINENEWLINEThe technique is to use a generalized Morse lemma, the Lebesgue density theorem and the Vitali covering lemma to transfer the problem to the sharp estimate of some formulas.