Tur\'an density of stars in uniformly dense hypergraphs

A $3$-uniform hypergraph (or $3$-graph) $H=(V,E)$ is $(d,\mu,1)$-\emph{dense} if for any subsets $X,Y,Z\subseteq V$, the number of triples $(x,y,z)\in X\times Y\times Z$ such that $\{x,y,z\}$ is an edge of $H$ is at least $d|X||Y||Z|-\mu |V|^3$. The \emph{$k$-star} $S_k$ is the $3$-graph with a center vertex and $k$ distinct leaf vertices, whose edge set consists of all triples containing the center and two distinct leaves. Restricting to $dot$-dense $3$-graphs, determining the \emph{$1$-uniform Tur\'an density} $\pi_1(S_k)$ of $S_k$ for $k\ge 4$ was proposed by Schacht in ICM 2022. In particular, Reiher, R\"odl and Schacht gave a palette construction showing that $\pi_1(S_k)\ge \frac{k^2-5k+7}{(k-1)^2}$ for $k\ge 3$, and also proved that $\pi_1(S_3)=1/4$. Lamaison and Wu later showed that this palette construction is optimal for $k\ge 48$. In this paper, we improve the results of Lamaison and Wu by proving that \[ \pi_1(S_k)=\frac{k^2-5k+7}{(k-1)^2} \qquad\text{for all } k\ge 9. \]