On polynomial progressions via transference

We prove new cases of reasonable bounds for the polynomial Szemerédi theorem both over $\mathbb{Z}\mathrm{/}N\mathbb{Z}$ with $N$ prime and over the integers. In particular, we prove reasonable bounds for Szemerédi’s theorem in the integers with fixed polynomial common difference. That is, we prove for any polynomial $P(y)\in \mathbb{Z}[y]$ with $P(0)=0$, that the largest subset $A\subseteq [N]$ avoiding the pattern $$x,x+P(y),\dots ,x+kP(y)$$ has size bounded by ${\ll }_{P,k}N(\log \log \log N{)}^{-{\mathrm{\Omega }}_{P,k}(1)}\mathrm{.}$