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

Abstract: 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 $k\ge3$ and $θ>17/30$, for all sufficiently large $X$ and all $x\in[X,2X]$, [ #{\text{$k$-APs of primes in }[x,x+x^θ]}\ \gg_{k,θ}\ \frac{N^{2}}{\big((\varphi(W)/W)^{k}(\log R)^{k}\big)}\ \asymp\ \frac{X^{2θ}}{(\log X)^{k+1+o(1)}}, ] where $W:=\prod_{p\le \tfrac12\log\log X}p$, $N:=\lfloor x^θ/W\rfloor$, 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 \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. The proofs in this paper were assisted by GPT-5.

• The paper establishes that intervals [x, x+x^θ] with θ > 17/30 consistently contain a positive fraction of the expected number of k-term prime arithmetic progressions.
• It leverages a uniform prime number theorem over short intervals combined with a modified Selberg/GPY sieve to ensure pseudorandom conditions for applying Szemerédi's theorem.
• The work extends combinatorial techniques and zero-density estimates, highlighting the integration of AI-assisted methods in advancing our understanding of prime distributions.

Arithmetic Progressions of Primes in Short Intervals Beyond the 17/30 Barrier

Introduction

The exploration of arithmetic progressions (APs) within the set of prime numbers is a classic inquiry rooted in number theory. Following the significant advancement by Green and Tao, who proved that prime numbers contain arbitrarily long APs, this paper extends the inquiry into the frequency and structure of $k$-term APs within short intervals of prime numbers. Specifically, the study demonstrates that for $\theta > 17/30$, sufficiently long intervals $[x, x + x^\theta]$ contain numerous $k$-APs of prime numbers. This result builds on recent advances in zero-density estimates and the Green-Tao transference principle.

Methodology

The method hinges on a uniform prime number theorem (PNT) over short intervals for exponents $\theta > 17/30$, derived from recent results by Guth and Maynard. This theorem is critical in establishing that the number of primes within such an interval is approximately the size of the interval, correcting for negligible errors as $X \to \infty$. To leverage this statement, a $W$-trick adaptation is applied to realign primes in congruence classes under the modulus $W$, facilitating the use of relative Szemerédi theorems to count $k$-APs within these intervals.

Of particular note is the role of the modified weights based on the Selberg/GPY sieve majorant, adapted to ensure pseudorandomness conditions necessary for the application of Szemerédi's theorem. This involves constructing a dense model supported by primes in short intervals, controlled by parameters $N$ and $R$, where $R = N^\eta$ and $\Lambda_R(m)$ is a weighted divisor sum truncating at $R$.

Key Results

The authors' primary theorem states that for fixed $k \ge 3$ and $\theta > 17/30$, every sufficiently large $X$ allows for intervals $[x, x+x^\theta]$ with $x \in [X, 2X]$ hosting at least a constant multiple of the expected number of $k$-term APs of primes derived from the model assumptions. The consistency and uniformity of this count are emphasized, reflecting minimal variance across translates within the defined interval.

A supplementary analysis extends this finding in a probabilistic setting, demonstrating that, for a positive proportion of starting points $x$, the interval $[x, x+x^\theta]$ conforms to the expected density of $k$-APs, discounting only an exceptional set of starting points with asymptotically negligible density.

Implications and Future Directions

This advancement opens several avenues for further research. Practically, it underscores the effectiveness of combining zero-density methods with advanced combinatorial techniques to analyze finely structured sets like the primes. Theoretically, it invites a closer examination of the parameters driving the distribution of primes in short intervals, specifically how these formulations might adapt and evolve as $\theta$ approaches the conjectured limits dictated by number theory arithmetic conjectures.

The utilization of AI-assisted methodologies, such as GPT-5 in proof-writing, signifies a novel trend where computational tools augment rigorous mathematical argumentation, potentially expediting the discovery of results that demand extensive computation and meticulous analysis.

Conclusion

In summary, the paper provides a significant mathematical bridge, escalating our understanding of prime distribution in arithmetic progressions within short, localized intervals. It sets the stage for expanding classical number-theoretic insights into more complex arrangements and increasingly granular levels, bolstered by contemporary computational techniques that propel forward the frontier of mathematical research.