Abstract: In my talk I am going to talk about how large are the level sets of a generic 1-Hölder-alpha real function in the sense of Hausdorff dimension and box dimension. Initially, we were interested in the former one, while our most recent proceedings concern the latter. To provide a better display of the differences, which are the most intriguing parts of this new direction, I aim to give a comparative overview of the two theories. Besides theorems and estimates valid for large families of fractals, I also present lower and upper estimates concerning the Sierpiński triangle, and explain what ideas they rely on. To conclude the talk, I will mention some open problems.

Based on joint papers with Zoltán Buczolich and Gáspár Vértesy.