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 of the above theorem uses ergodic theory, which makes it difficult to extract quantitative bounds from it; and standard methods of “finitizing” ergodic theory results, for instance by replacing Host–Kra seminorms by their Gowers uniformity norm counterparts, run into a number of technical difficulties here due to the need to work with multiple scales due to the presence of polynomials, as well as the fact that many of the conjectural uniformity properties of the prime numbers at small scales remain unproven.

Nevertheless, we are able to get reasonable quantitative results (with density bounds that are roughly single or doubly logarithmic in scale) in the following special cases:

• linear polynomials;
• polynomials of distinct degree; and
• multiples of a fixed polynomial.

The precise statements are slightly technical and not reproduced here.

Previous quantitative bounds in the linear case were obtained by Leng and by Teräväinen and myself, but our new method improves upon these bounds by roughly one iterated logarithm. On the other hand, in the case of two term progressions of spacing , there is a much stronger quantitative result (with polynomial dependence of constants) due to Green; our method do not recover that result.

Our methods use a variety of old and new methods in the subject. For instance, we use the (now quite standard) “-trick” to restrict the primes to a single congruence class to improve their uniformity properties (which, thanks to the recent work of Leng and Matthiesen–Teräväinen–Wang, are now quite strong quantitatively); and for good configurations of configurations, one can also use a recent transference theorem of Altman and Sawhney to compare the polynomial averages with simpler linear ones, without having to pass to short scales. In order to get relatively strong bounds unconditionally, a “Siegel approximation” for the primes is used taking into account the potential influence of a Siegel zero. It will not be surprising to the experts that quantitative inverse Gowers theorems and nilsequence equidistribution theorems also play a major role.

A key technical difficulty is the presence of the modulus in the coefficients of the polynomial progressions after changes of variable, which forced us to make several of the existing estimates uniform over such coefficients (assuming they are not unreasonably large).This caused several complications that made a correct argument to more time-consuming to locate than initially planned.

AI usage in this work was fairly light, being restricted to proofreading and literature search only.