Quantitative bounds for sets lacking polynomial progressions with shifted prime difference

Abstract

We prove quantitative polynomial Szemer\'edi-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.