On the algebraic and topological structure of the set of Tur\'an densities

The present paper is concerned with the various algebraic structures supported by the set of Tur\'an densities. We prove that the set of Tur\'an densities of finite families of r-graphs is a non-trivial commutative semigroup, and as a consequence we construct explicit irrational densities for any r >= 3. The proof relies on a technique recently developed by Pikhurko. We also show that the set of all Tur\'an densities forms a graded ring, and from this we obtain a short proof of a theorem of Peng on jumps of hypergraphs. Finally, we prove that the set of Tur\'an densities of families of r-graphs has positive Lebesgue measure if and only if it contains an open interval. This is a simple consequence of Steinhaus's theorem.