Investigation of smooth functions and analytic sets using fractal dimensions

D'Aniello, Emma

Bollettino della Unione matematica italiana, Série 8, 7B (2004) no. 3, pp. 637-646

Résumé

We start from the following problem: given a function $f \colon [0, 1] \to [0, 1]$ what can be said about the set of points in the range where level sets are «big» according to an opportune definition. This yields the necessity of an analysis of the structure of level sets of $C^{n}$ functions. We investigate the analogous problem for $C^{n, a}$ functions. These are in a certain way intermediate between $C^{n}$ and $C^{n+1}$ functions. The results involve a mixture of Real Analysis, Geometric Measure Theory and Classical Descriptive Set Theory.