Quantitative bounds for sets lacking polynomial progressions with shifted prime difference

Ben Krause, Hamed Mousavi, Joni Teräiväinen, and I have just uploaded to the arXiv our paper Quantitative bounds for sets lacking polynomial progressions with shifted prime difference. The purpose of this paper is to obtain quantative versions of this theorem of Wooley and Ziegler: Theorem 1 Let be polynomials of one variable with integer coefficients with zero constant term, and let be a set of integers of positive density. Then there exist infinitely many primes such that contains a progression of the form for some integer . This generalizes the famous theorem of Szemerédi in two ways: firstly, by considering “polynomial progressions” instead of arithmetic progressions, and secondly by requiring the shift parameter to be one less than a prime . The first extension of Szemerédi’s theorem is a theorem of Bergelson and Leibman; and the second extension is also obtainable by combining the arguments of Frantzikinakis, Host and Kra with the results of Green, Ziegler, and myself. The proof…