Supersaturation of odd linear cycles

An $r$-uniform linear cycle of length $\ell$, denoted by $C^r_{\ell}$, is an $r$-graph with $\ell$ edges $e_1,e_2,\dots,e_{\ell}$ where $e_i=\{v_{(r-1)(i-1)},v_{(r-1)(i-1)+1},\dots,v_{(r-1)i}\}$ (here $v_0=v_{(r-1)\ell}$). For $0<\delta<1$ and $n$ sufficiently large, we show that every $n$-vertex $r$-graph $G$ with $n^{r-\delta}$ edges contains at least $n^{(r-1)(2\ell+1)-\delta(2\ell+1+\frac{4\ell-1}{(r-1)(2\ell+1)-3})-o(1)}$ copies of $C^r_{2\ell+1}$. Further, conditioning on the existence of dense high-girth hypergraphs, we show that there exists $n$-vertex $r$-graphs with $n^{r-\delta}$ edges and at most $n^{(r-1)(2\ell+1)-\delta(2\ell+1+\frac{1}{(r-1)\ell-1})+o(1)}$ copies of $C^r_{2\ell+1}$.