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 k3 and θ>17/30, for all sufficiently large X and all x, 2X, \ \#\k-APs of primes in [x, x+x^θ\\ ₊, ⏗\ N^2 ( (φ (W) /W) ^{k (R) ^k) }\ \ X^2θ (X) ^{k+1+o (1) }, \] where W: = ₁₂ ₗp, 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 (x) ^, 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.
Le Duc Hieu (Fri,) studied this question.