Minimum shadows in uniform hypergraphs and a generalization of the Takagi function

The shadow function is closely related to the Kruskal-Katona Theorem. The Takagi function is a standard example of a nowhere differentiable continuous function. The purpose of this paper is to exhibit a rather surprising relationship between the shadow function and the Takagi function. Using this relationship, one can approximately compute the size of minimum shadows in uniform hypergraphs with a given number of edges. In order to describe the asymptotic behaviour of the size of shadows, we introduce a new, generalized Takagi function. The results explain the difficulties, often encountered when using the best possible bounds arising from the Kruskal-Katona Theorem.