Abstract
For a set of primes
for some
Funding source: Suomen Akatemia
Award Identifier / Grant number: 137883
Award Identifier / Grant number: 138522
Funding statement: Kaisa Matomäki was supported by Academy of Finland grants no. 137883 and 138522. Xuancheng Shao was supported by a Glasstone Research Fellowship.
Acknowledgements
The present work started when both authors were visiting CRM in Montreal during the analytic part of the thematic year in number theory in Fall 2014, whose hospitality is greatly appreciated. Thanks to Ben Green for helpful discussions. Thanks to the anonymous referee for valuable comments and suggestions.
References
[1] D. Bleichenbacher, The continuous postage stamp problem, unpublished manuscript (2003). Search in Google Scholar
[2] J. Bourgain, Estimates related to sumfree subsets of sets of integers, Israel J. Math. 97 (1997), 71–92. 10.1007/BF02774027Search in Google Scholar
[3] J. W. S. Cassels, Measures of the non-convexity of sets and the Shapley–Folkman–Starr theorem, Math. Proc. Cambridge Philos. Soc. 78 (1975), no. 3, 433–436. 10.1017/S0305004100051884Search in Google Scholar
[4] I. Ekeland and R. Temam, Convex analysis and variational problems, Stud. Math. Appl. 1, North-Holland, Amsterdam 1976. Search in Google Scholar
[5] J. Friedlander and H. Iwaniec, Opera de cribro, Amer. Math. Soc. Colloq. Publ. 57, American Mathematical Society, Providence 2010. 10.1090/coll/057Search in Google Scholar
[6] A. Granville, D. Koukoulopoulos and K. Matomäki, When the sieve works, Duke Math. J. 164 (2015), no. 10, 1935–1969. 10.1215/00127094-3120891Search in Google Scholar
[7] B. Green, A Szemerédi-type regularity lemma in abelian groups, with applications, Geom. Funct. Anal. 15 (2005), no. 2, 340–376. 10.1007/s00039-005-0509-8Search in Google Scholar
[8] B. Green and I. Z. Ruzsa, Freiman’s theorem in an arbitrary abelian group, J. Lond. Math. Soc. (2) 75 (2007), no. 1, 163–175. 10.1112/jlms/jdl021Search in Google Scholar
[9] H. Iwaniec and E. Kowalski, Analytic number theory, Amer. Math. Soc. Colloq. Publ. 53, American Mathematical Society, Providence 2004. 10.1090/coll/053Search in Google Scholar
[10] D. Král, O. Serra and L. Vena, A combinatorial proof of the removal lemma for groups, J. Combin. Theory Ser. A 116 (2009), no. 4, 971–978. 10.1016/j.jcta.2008.12.003Search in Google Scholar
[11] H. W. Lenstra, Jr. and C. Pomerance, Primality testing with Gaussian periods, preprint (2011), http://www.math.dartmouth.edu/~carlp/aks06-2015.pdf. 10.4171/JEMS/861Search in Google Scholar
[12] I. Z. Ruzsa, The Brunn–Minkowski inequality and nonconvex sets, Geom. Dedicata 67 (1997), no. 3, 337–348. 10.1023/A:1004958110076Search in Google Scholar
[13] A. Shapira, A proof of Green’s conjecture regarding the removal properties of sets of linear equations, J. Lond. Math. Soc. (2) 81 (2010), no. 2, 355–373. 10.1112/jlms/jdp076Search in Google Scholar
[14] B. Szegedy, The symmetry preserving removal lemma, Proc. Amer. Math. Soc. 138 (2010), no. 2, 405–408. 10.1090/S0002-9939-09-09860-8Search in Google Scholar
[15] E. Szemerédi and V. H. Vu, Finite and infinite arithmetic progressions in sumsets, Ann. of Math. (2) 163 (2006), no. 1, 1–35. 10.4007/annals.2006.163.1Search in Google Scholar
[16] T. Tao and V. H. Vu, John-type theorems for generalized arithmetic progressions and iterated sumsets, Adv. Math. 219 (2008), no. 2, 428–449. 10.1016/j.aim.2008.05.002Search in Google Scholar
[17] T. Tao and V. H. Vu, Additive combinatorics, Cambridge Stud. Adv. Math. 105, Cambridge University Press, Cambridge 2010. Search in Google Scholar
[18] J. Wolf, The structure of popular difference sets, Israel J. Math. 179 (2010), 253–278. 10.1007/s11856-010-0081-2Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- When the sieve works II
- A gluing approach for the fractional Yamabe problem with isolated singularities
- Dehn functions and Hölder extensions in asymptotic cones
- Gap theorem on Kähler manifolds with nonnegative orthogonal bisectional curvature
- Kähler geometry of horosymmetric varieties, and application to Mabuchi’s K-energy functional
- Asymptotics for the level set equation near a maximum
- Construction of constant mean curvature n-noids using the DPW method
- Irreducible components of the eigencurve of finite degree are finite over the weight space
- Structure theorems for singular minimal laminations
Articles in the same Issue
- Frontmatter
- When the sieve works II
- A gluing approach for the fractional Yamabe problem with isolated singularities
- Dehn functions and Hölder extensions in asymptotic cones
- Gap theorem on Kähler manifolds with nonnegative orthogonal bisectional curvature
- Kähler geometry of horosymmetric varieties, and application to Mabuchi’s K-energy functional
- Asymptotics for the level set equation near a maximum
- Construction of constant mean curvature n-noids using the DPW method
- Irreducible components of the eigencurve of finite degree are finite over the weight space
- Structure theorems for singular minimal laminations