Improved bounds for five-term arithmetic progressions

Let $r_5(N)$ be the largest cardinality of a set in $\{1,\dots ,N\}$ which does not contain $5$ elements in arithmetic progression. Then there exists a constant $c\in (0,1)$ such that $$r_5(N)\ll \frac{N}{\exp ((\log \log N{)}^c)}\mathrm{.}$$ Our work is a consequence of recent improved bounds on the $U^4$-inverse theorem of the first author and the fact that $3$-step nilsequences may be approximated by locally cubic functions on shifted Bohr sets. This combined with the density increment strategy of Heath-Brown and Szemer{é}di, codified by Green and Tao, gives the desired result.