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The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Product of difference sets of the set of primes
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by Sayan Goswami;

Proc. Amer. Math. Soc. 151 (2023), 5081-5086

DOI: https://doi.org/10.1090/proc/16478

Published electronically: September 20, 2023

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Abstract:

In a recent work, A. Fish [Proc. Amer. Math. Soc. 146 (2018), pp. 3449–3453] proved that if $E_{1}$ and $E_{2}$ are two subsets of $\mathbb {Z}$ of positive upper Banach density, then there exists $k\in \mathbb {Z}$ such that $k\cdot \mathbb {Z}\subset \left (E_{1}-E_{1}\right )\cdot \left (E_{2}-E_{2}\right ).$ In this article we will show that a similar result is true for the set of primes $\mathbb {P}$ (which has density $0$). We will prove that there exists $k\in \mathbb {N}$ such that $k\cdot \mathbb {N}\subset \left (\mathbb {P}-\mathbb {P}\right )\cdot \left (\mathbb {P}-\mathbb {P}\right ),$ where $\mathbb {P}-\mathbb {P}=\left \{ p-q:p>q\text { and }p,q\in \mathbb {P}\right \} .$

References

V. Bergelson, A. Blass, M. Di Nasso and R. Jin, Ultrafilters across Mathematics International Congress ULTRAMATH 2008: Applications of Ultrafilters and Ultraproducts in Mathematics June 1–7, 2008 Pisa, Italy.
Michael Björklund and Alexander Fish, Characteristic polynomial patterns in difference sets of matrices, Bull. Lond. Math. Soc. 48 (2016), no. 2, 300–308. MR 3483067, DOI 10.1112/blms/bdw008
Robert Ellis, Distal transformation groups, Pacific J. Math. 8 (1958), 401–405. MR 101283, DOI 10.2140/pjm.1958.8.401
P. Erdős and E. Szemerédi, On sums and products of integers, Studies in pure mathematics, Birkhäuser, Basel, 1983, pp. 213–218. MR 820223
Alexander Fish, On product of difference sets for sets of positive density, Proc. Amer. Math. Soc. 146 (2018), no. 8, 3449–3453. MR 3803669, DOI 10.1090/proc/14078
H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory, Princeton University Press, Princeton, NJ, 1981. M. B. Porter Lectures. MR 603625, DOI 10.1515/9781400855162
Daniel A. Goldston, János Pintz, and Cem Y. Yıldırım, Primes in tuples. I, Ann. of Math. (2) 170 (2009), no. 2, 819–862. MR 2552109, DOI 10.4007/annals.2009.170.819
Wen Huang and XiaoSheng Wu, On the set of the difference of primes, Proc. Amer. Math. Soc. 145 (2017), no. 9, 3787–3793. MR 3665033, DOI 10.1090/proc/13533
Neil Hindman and Dona Strauss, Algebra in the Stone-Čech compactification, De Gruyter Textbook, Walter de Gruyter & Co., Berlin, 2012. Theory and applications; Second revised and extended edition [of MR1642231]. MR 2893605
S. V. Konyagin and I. D. Shkredov, New results on sums and products in $\Bbb {R}$, Tr. Mat. Inst. Steklova 294 (2016), no. Sovremennye Problemy Matematiki, Mekhaniki i Matematicheskoĭ Fiziki. II, 87–98 (Russian, with Russian summary). English version published in Proc. Steklov Inst. Math. 294 (2016), no. 1, 78–88. MR 3628494, DOI 10.1134/S0371968516030055
L. Kronecker, Vorlesungenüber Zahlentheorie, I., Teubner, Leipzig, 1901, p. 68.
E. Maillet, L’intermédiaire des math, 12 (1905), 108.
James Maynard, Small gaps between primes, Ann. of Math. (2) 181 (2015), no. 1, 383–413. MR 3272929, DOI 10.4007/annals.2015.181.1.7
A. de. Polignac, Recherches nouvelles sur les nombres premiers, C. R. Acad. Sci. Paris 29 (1849), 397–401, Rectification: ibid. pp. 738–739.
D. H. J. Polymath, Variants of the Selberg sieve, and bounded intervals containing many primes, Res. Math. Sci. 1 (2014), Art. 12, 83. MR 3373710, DOI 10.1186/s40687-014-0012-7
Yitang Zhang, Bounded gaps between primes, Ann. of Math. (2) 179 (2014), no. 3, 1121–1174. MR 3171761, DOI 10.4007/annals.2014.179.3.7