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

We show that once $θ>17/30$, every sufficiently long interval [x,x+x^θ] contains many k-term arithmetic progressions of primes, uniformly in the starting point x. More precisely, for each fixed k≥3 and $θ>17/30$, for all sufficiently large X and all x∈[X,2X], \[ #\{k-APs of primes in [x,x+x^θ]\}\ k,θ\ {N²}{((φ(W)/W)ᵏ(log R)ᵏ)}\ \ {X2θ}{(log X)ᵏ⁺¹⁺ᵒ⁽¹⁾}, \] where W:=∏p≤ 12loglog Xp, N:= x^θ/W, and R:=N^η for a small fixed $η=η(k,θ)>0$. This is obtained by combining the uniform short-interval prime number theorem at exponents $θ>17/30$ (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 restricted to a fixed congruence class in tiny intervals (log x)^ε, 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.