On Whitney sets and their generalization (Q1772958)

Let \( h:[0,+\infty[\rightarrow [0,+\infty[ \) be an increasing function with \( h(0)=0 \). Say that a connected set \(H\subset {\mathbb R}^n \) is a \( W^{(h)} \)-set if there is a non-constant function \( f : H\rightarrow {\mathbb R} \) such that the condition \( \lim_{x\to x_0, x\in H} | f(x)-f(x_0)| /h(\| x-x_0\|)=0 \) holds for all \( x_0\in H \). A connected subset of \({\mathbb R}^n\) is called a \( C^{(h)} \)-set if it is not a \( W^{(h)} \)-set. The main theorem of the paper can then be stated as follows. Consider a curve \( \varphi:[\alpha,\beta]\rightarrow {\mathbb R}^{n+k} \), where \( k \) is a given integer, \( k\geq 1 \). Put \( E=\{t\in[\alpha,\beta[\, :\lim_{s\to t^+}\| \varphi(s)-\varphi(t)\|^k/| s-t| =\infty\} \), and assume that for all \( t_0\in E\), there are real numbers \( \delta\in ]0,\beta-t_0[ \), \( M>0 \), and \( n \) integers \( 1\leq i_1<\cdots< i_n\leq n+k \) such that the inequality \(| \varphi_{i_l}(t)-\varphi_{i_l}(t_0)|\leq M| t-t_0|^{1/k} \) (\(l=1,\ldots,n\)) holds for all \( t\in ]t_0,t_0+\delta[ \). Then \( \varphi([\alpha,\beta]) \) is a \( C^{(h)} \)-set for \( h(t)=t^k \). The proof is partly based on a slight generalization of a result of \textit{M. Laczkovich} and \textit{G. Petruska} [Real Anal. Exch. 10(1984/85), 313--323 (1985; Zbl 0593.26007)] concerning the case \( h(t)=t \).