Generating non-jumps from a known one

Let $r\ge 2$ be an integer. The real number $α\in [0,1]$ is a jump for $r$ if there exists a constant $c > 0$ such that for any $ε>0$ and any integer $m \geq r$, there exists an integer $n_0(ε, m)$ satisfying any $r$-uniform graph with $n\ge n_0(ε, m)$ vertices and density at least $α+ε$ contains a subgraph with $m$ vertices and density at least $α+c$. A result of Erdős, Stone and Simonovits implies that every $α\in [0,1)$ is a jump for $r=2$. Erdős asked whether the same is true for $r\ge 3$. Frankl and Rödl gave a negative answer by showing that $1-\frac{1}{l^{r-1}}$ is not a jump for $r$ if $r\ge 3$ and $l>2r$. After that, more non-jumps are found using a method of Frankl and Rödl. In this note, we show a method to construct maps $f \colon [0,1] \to [0,1]$ that preserve non-jumps, if $α$ is a non-jump for $r$ given by the method of Frankl and Rödl, then $f(α)$ is also a non-jump for $r$. We use these maps to study hypergraph Turán densities and answer a question posed by Grosu.

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