具有移位素数差的集合的多项式进展定量界
Quantitative bounds for sets lacking polynomial progressions with shifted prime difference
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中文总结 AI 辅助
该研究证明了移位参数限制在移位素数集合的多项式进展的定量Szemerédi型定理，改进了线性情形结果，不同次数非线性构型的定量界为首创，结合了整数多项式构型与素数Gowers一致性界等方法。
中文摘要 AI 辅助
我们证明了涉及移位参数限制在移位素数集合$\boldsymbol{P}-1$内的多项式进展的定量多项式Szemerédi型定理。所涵盖的构型类型包括不同次数的进展以及涉及固定多项式整数倍的进展。对于长度至少为3的非线性构型，这些结果提供了此类定理的首个定量版本。在线性情形下，我们的结果改进了后两位作者的工作。我们的密度界在不同次数多项式情形下最强，其中它们给出了多重对数界，与Shao和Wang近期带整数移位的界形式相同。证明结合了整数中多项式构型的近期定量结果与素数的定量Gowers一致性界。对于固定多项式的倍数，我们采用Altman和Sawhney的比较论证，以获得通过W技巧生成的多项式族上的一致性。对于不同次数的进展，我们在整个密度增量论证中建立了素数加权与非加权多项式计数之间的一致性比较，并考虑了可能的Siegel零点。
英文摘要
We prove quantitative polynomial Szemerédi-type theorems involving polynomial progressions with shift parameter restricted to the set of shifted primes $\mathbb{P}-1$. The types of configurations covered are distinct degree progressions and progressions involving integer multiples of a fixed polynomial. For nonlinear configurations of length at least three, these results provide the first quantitative versions of such theorems. In the linear case, our results improve on work by the last two authors. Our density bounds are strongest in the case of distinct degree polynomials, where they give polylogarithmic bounds, of the same shape as recent bounds by Shao and Wang with integer shifts. The proofs combine recent quantitative results for polynomial configurations in the integers with quantitative Gowers uniformity bounds of the primes. For multiples of a fixed polynomial, we adapt a comparison argument of Altman and Sawhney to obtain uniformity over the polynomial families produced by the $W$-trick. For distinct degree progressions, we establish a comparison between prime-weighted and unweighted polynomial counts that is uniform throughout the density increment argument and accounts for a possible Siegel zero.
