Intervals of hypergraph Turán densities

We prove that, for every integer $r\ge 3$, the set ${\mathrm{\Pi }}_{\mathrm{\infty }}^{(r)}$ of Turán densities of (possibly infinite) families of $r$-graphs contains non-degenerate intervals, including an interval of the form $[1-{\delta }_r,1]$ for some ${\delta }_r>0$. This answers a question of Frankl, Peng, Rödl and Talbot from 2007. This also shows that the Hausdorff dimension of ${\mathrm{\Pi }}_{\mathrm{\infty }}^{(r)}$ has the maximum possible value 1, thus resolving a question of Grosu from 2016, whereas previously it was not even known whether it is non-zero. We also derive that the set of uniform Turán densities of finite families of $3$-graphs is dense in a non-degenerate interval.