Arithmetic progressions of primes in short intervals beyond the 17/30 barrier

We show that once $\theta >17\mathrm{/}30$, every sufficiently long interval $[x,x+x^{\theta }]$ contains many $k$-term arithmetic progressions of primes, uniformly in the starting point $x$. More precisely, for each fixed $k\ge 3$ and $\theta >17\mathrm{/}30$, for all sufficiently large $X$ and all $x\in [X,2X]$, $$\mathrm{\#}\{k\text{-APs of primes in }[x,x+x^{\theta }]\}{\gg }_{k,\theta }\frac{N^2}{((\phi (W)\mathrm{/}W{)}^k(\log R{)}^k)}\asymp \frac{X^{2\theta }}{(\log X{)}^{k+1+o(1)}},$$ where $W:={\prod }_{p\le \frac{1}{2}\log \log X}p$, $N:=\lfloor x^{\theta }\mathrm{/}W\rfloor$, and $R:=N^{\eta }$ for a small fixed $\eta =\eta (k,\theta )>0$. This is obtained by combining the uniform short-interval prime number theorem at exponents $\theta >17\mathrm{/}30$ (a consequence of recent zero-density estimates of Guth and Maynard) with the Green-Tao transference principle (in the relative Szemerédi form) on a window-aligned $W$-tricked block. We also record a concise Maynard-type lemma on dense clusters \emph{restricted to a fixed congruence class} in tiny intervals $(\log x{)}^{\epsilon }$, which we use as a warm-up and for context. An appendix contains a short-interval Barban-Davenport-Halberstam mean square bound (uniform in $x$) that we use as a black box for variance estimates. The proofs in this paper were assisted by GPT-5.