longer progressions $$\label{longer}
x,\ x+ P_1(y), \ \dots, \ x + P_m(y),$$ where the $P_i \in {\mathbb{Z}}[y]$ have zero constant term and $\deg P_1 < \dots < \deg P_m$.

As was mentioned above, the key input to this paper is the local $U^1$-control result [@PelusePrendivilleQuantitative Theorem 7.1]. Replacing this with [@PeluseBounds Theorem 3.3], our argument generalises in a straightforward manner to yield polylogarithmic bounds for subsets of $\{1,\dots,N\}$ lacking when $m = 2$, that is, for all three-term polynomial progressions with distinct degrees and zero constant term.

Obtaining polylogarithmic bounds for longer polynomial progressions requires an additional idea. We sketch a strategy in §\[longer section\], which relies on obtaining an appropriate generalisation of [@PeluseBounds Theorem 3.3], a generalisation that would require re-running the majority of the arguments therein.

Acknowledgements {#acknowledgements .unnumbered}
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S. Peluse is supported by the NSF Mathematical Sciences Postdoctoral Research Fellowship Program under Grant No. DMS-1903038

An outline of our argument {#sec3}
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Effective Szemerédi-type theorems are commonly proved via a density increment strategy, the prototypical example being the proof of Roth’s theorem [@RothCertainI] on three-term arithmetic progressions. This strategy begins with a set $A \subset \{1, \dots, N\}$ of density $\delta := |A|/N$ that lacks the 