Abstract
In a previous paper of the authors, we showed that for any polynomials \(P_{1},\ldots,P_{k} \in \mathbb{Z}[\mathbf{m}]\) with \(P_{1}(0) =\ldots = P_{k}(0)\) and any subset A of the primes in [N] = { 1, …, N} of relative density at least δ > 0, one can find a “polynomial progression” \(a + P_{1}(r),\ldots,a + P_{k}(r)\) in A with 0 < | r | ≤ N o(1), if N is sufficiently large depending on \(k,P_{1},\ldots,P_{k}\) and δ. In this paper we shorten the size of this progression to \(0 < \vert r\vert \leq \log ^{L}N\), where L depends on \(k,P_{1},\ldots,P_{k}\) and δ. In the linear case P i = (i − 1)m, we can take L independent of δ. The main new ingredient is the use of the densification method of Conlon, Fox, and Zhao to avoid having to directly correlate the enveloping sieve with dual functions of unbounded functions.
To Helmut Maier on his 60th birthday
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Notes
- 1.
All sums and products over p in this paper are understood to be ranging over primes.
- 2.
The “dual function condition” that the dual function of the enveloping sieve is bounded already failed in the arguments in [19], which was a significant cause of the complexity of that paper due to the need to find substitutes for this condition (in particular, the correlation condition became significantly more difficult to even state, let alone prove). But the arguments in [6] do not require any version of the dual function condition at all, leading to some simplifications in the current argument over those in [19].
- 3.
Actually, the weaker lower bound \(\pi (x) \gg \frac{x} {\log x}\) of Chebyshev would suffice here.
- 4.
The inradius of a convex body is the radius of the largest open ball one can inscribe inside the body.
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Acknowledgements
The first author is supported by a Simons Investigator grant, the James and Carol Collins Chair, the Mathematical Analysis & Application Research Fund Endowment, and by NSF grant DMS-1266164. Part of this research was performed while the first author was visiting the Institute for Pure and Applied Mathematics (IPAM), which is supported by the National Science Foundation. The second author is supported by ISF grant 407/12. The second author was on sabbatical at Stanford while part of this work was carried out; she would like to thank the Stanford math department for its hospitality and support. Finally, the authors thank the anonymous referee for a careful reading of the paper.
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Tao, T., Ziegler, T. (2015). Narrow Progressions in the Primes. In: Pomerance, C., Rassias, M. (eds) Analytic Number Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-22240-0_22
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