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Dimensions of level sets of generic continuous functions and generic H\"olderfunctions defined on a fractal $F$ encode information about the geometry, ``thethickness" of $F$. While in the continuous case this quantity is related to areasonably tame dimension notion which is called the topological Hausdorffdimension of $F$, the H\"older case seems to be highly nontrivial. A number ofearlier papers attempted to deal with this problem, carrying out investigationin the case of Hausdorff dimension and box dimension. In this paper we continueour study of the Hausdorff dimension of almost every level set of generic$1$-H\"older-$ {\alpha}$ functions, denoted by $D_{*}( {\alpha}, F)$. Wesubstantially improve previous lower and upper bounds on $D_{*}( {\alpha},\Delta)$, where $\Delta$ is the Sierpi\'nski triangle, achieving asymptoticallyequal bounds as $\alpha\to 0+$. Using a similar argument, we also give an evenstronger lower bound on the generic lower box dimension of level sets. Finally,we construct a connected fractal $F$ on which there is a phase transition of$D_{*}( {\alpha}, F)$, thus providing the first example exhibiting thisbehavior.

Better estimates of H\"older thickness of fractals