Quantitative bounds for sets lacking polynomial progressions with shifted prime difference

per Terence Tao, along with collaborators Ben Krause, Hamed Mousavi, Joni Teräiväinen and himself, have just uploaded a new preprint to the arXiv in August twenty twenty-six. The paper titled Quantitative bounds for sets lacking polynomial progressions with shifted prime difference, refines a theorem of Wooley and Ziegler. It shows that for any set of positive density and certain integer‑coefficient polynomials with zero constant term, there are infinitely many primes forming a progression of the form k, k+p, … where the shift equals a prime minus one. This extends Szemerédi’s theorem by using polynomial progressions and a prime shift, and it pushes quantitative bounds forward, improving earlier results by roughly one iterated logarithm in the linear case. The authors employ ergodic theory, uniformity norms and recent advances in additive combinatorics to obtain these estimates. The preprint is now available for download on the arXiv server.

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