Improved bounds for five-term arithmetic progressions

Let $r_5(N)$ be the largest cardinality of a set in $\{1,\ldots,N\}$ which does not contain 5 elements in arithmetic progression. Then there exists a constant $c\in (0,1)$ such that \[r_5(N)\ll \frac{N}{\exp\!((\!\log\log N)^{c})}.\] Our work is a consequence of recent improved bounds on the $U^4$ -inverse theorem of J. Leng 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 Szemerédi, codified by Green and Tao, gives the desired result.