= Polynomial-length progression in a dense triple sumset
{title2=$|P|\geq N^{r_\delta},\quad r_\delta>0$}
For fixed positive density $\delta$, every sufficiently large $A\subseteq[1,N]$ of size at least $\delta N$ has a long <arithmetic progression> in its triple <sumset>. Embed it in $\mathbb Z/(8N)\mathbb Z$, where its density is at least $\delta/8$. A <translated Bohr neighborhood in a triple sumset> has bounded rank and fixed positive radius depending only on $\delta$. A torus <pigeonhole principle> argument supplies a nonzero step whose small multiples stay in the <Bohr set>, with polynomially many multiples. All resulting residues have representatives in $[3,3N]$, an interval shorter than half the modulus, so their successive ordinary integer differences agree and give an actual integer <arithmetic progression>. The positive constant prefactor is absorbed by reducing the exponent for sufficiently large $N$.
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