📝 SummarXiv

Abstract

We show that once $\theta>17/30$, 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 $k\ge3$ and $\theta>17/30$, for all sufficiently large $X$ and all $x\in[X,2X]$, \[ \#\{\text{$k$-APs of primes in }[x,x+x^\theta]\}\ \gg_{k,\theta}\ \frac{N^{2}}{\big((\varphi(W)/W)^{k}(\log R)^{k}\big)}\ \asymp\ \frac{X^{2\theta}}{(\log X)^{k+1+o(1)}}, \] where $W:=\prod_{p\le \tfrac12\log\log X}p$, $N:=\lfloor x^\theta/W\rfloor$, 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 $\theta>17/30$ (a consequence of recent zero-density estimates of Guth and Maynard) with the Green-Tao transference principle (in the relative Szemer\'edi 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 $(\log x)^\varepsilon$, 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.