Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-848d4c4894-jbqgn Total loading time: 0 Render date: 2024-06-16T09:29:03.590Z Has data issue: false hasContentIssue false

Finite Field Models in Arithmetic Combinatorics – Twenty Years On

Published online by Cambridge University Press:  23 May 2024

By
Sarah Peluse
Edited by
Felix Fischer  and
Robert Johnson
Felix Fischer
Affiliation:
Queen Mary University of London
Robert Johnson
Affiliation:
Queen Mary University of London
Get access

Summary

About twenty years ago, Green wrote a survey article on the utility of looking at toy versions over finite fields of problems in additive combinatorics. This survey was extremely influential, and the rapid development of additive combinatorics necessitated a follow-up article ten years later, which was written by Wolf. Since the publication of Wolf’s article, an immense amount of progress has been made on several central open problems in additive combinatorics in both the finite field model and integer settings. This survey covers some of the most significant results of the past ten years and suggests future directions.

Type
Chapter
Information
Surveys in Combinatorics 2024 , pp. 159 - 200
Publisher: Cambridge University Press
Print publication year: 2024

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alon, N. and Dubiner, M., Zero-sum sets of prescribed size, Combinatorics, Paul Erdős is eighty, Vol. 1, Bolyai Soc. Math. Stud., János Bolyai Math. Soc., Budapest, 1993, pp. 33–50. MR 1249703Google Scholar
Altman, D., On a conjecture of Gowers and Wolf, Discrete Anal. (2022), Paper No. 10, 13. MR 4481407Google Scholar
Arala, N., A maximal extension of the Bloom-Maynard bound for sets with no square differences, preprint (2023), arXiv:2303.03345.Google Scholar
Balog, A., Pelikán, J., Pintz, J., and Szemerédi, E., Difference sets without κth powers, Acta Math. Hungar. 65 (1994), no. 2, 165–187. MR 1278767CrossRefGoogle Scholar
Bateman, M. and Katz, N. H., New bounds on cap sets, J. Amer. Math. Soc. 25 (2012), no. 2, 585–613. MR 2869028CrossRefGoogle Scholar
Behrend, F. A., On sets of integers which contain no three terms in arithmetical progression, Proc. Nat. Acad. Sci. U. S. A. 32 (1946), 331–332. MR 0018694CrossRefGoogle ScholarPubMed
Bergelson, V. and Leibman, A., Polynomial extensions of van der Waerden’s and Szemerédi’s theorems, J. Amer. Math. Soc. 9 (1996), no. 3, 725–753. MR 1325795CrossRefGoogle Scholar
Bergelson, V., Leibman, A., and Lesigne, E., Complexities of finite families of polynomials, Weyl systems, and constructions in combinatorial number theory, J. Anal. Math. 103 (2007), 47–92. MR 2373264Google Scholar
Bergelson, V., Tao, T., and Ziegler, T., An inverse theorem for the uniformity seminorms associated with the action of Fp∞, Geom. Funct. Anal. 19 (2010), no. 6, 1539–1596. MR 2594614CrossRefGoogle Scholar
Blasiak, J., Church, T., Cohn, H., Grochow, J. A., Naslund, E., F. Sawin, W., and Umans, C., On cap sets and the group-theoretic approach to matrix multiplication, Discrete Anal. (2017), Paper No. 3, 27. MR 3631613Google Scholar
Bloom, T. F., A quantitative improvement for Roth’s theorem on arithmetic progressions, J. Lond. Math. Soc. (2) 93 (2016), no. 3, 643–663. MR 3509957CrossRefGoogle Scholar
Bloom, T. F. and Maynard, J., A new upper bound for sets with no square differences, Compos. Math. 158 (2022), no. 8, 1777–1798. MR 4490931CrossRefGoogle Scholar
Bloom, T. F. and Sisask, O., Breaking the logarithmic barrier in Roth’s theorem on arithmetic progressions, preprint (2020), arXiv:2007.03528.Google Scholar
Bloom, T. F. and Sisask, O., An improvement to the Kelley-Meka bounds on three-term arithmetic progressions, preprint (2023), arXiv:2309.02353.Google Scholar
Bloom, T. F. and Sisask, O., The Kelley–Meka bounds for sets free of three-term arithmetic progressions, preprint (2023), arXiv:2302.07211.Google Scholar
Bourgain, J., On triples in arithmetic progression, Geom. Funct. Anal. 9 (1999), no. 5, 968–984. MR 1726234CrossRefGoogle Scholar
Bourgain, J., Roth’s theorem on progressions revisited, J. Anal. Math. 104 (2008), 155–192. MR 2403433CrossRefGoogle Scholar
Bourgain, J. and Chang, M.-C., Nonlinear Roth type theorems in finite fields, Israel J. Math. (2017).CrossRefGoogle Scholar
Brown, T. C. and Buhler, J. P., A density version of a geometric Ramsey theorem, J. Combin. Theory Ser. A 32 (1982), no. 1, 20–34. MR 640624CrossRefGoogle Scholar
Croot, E., Lev, V. F., and Pach, P. P., Progression-free sets in Zn4 are exponentially small, Ann. of Math. (2) 185 (2017), no. 1, 331–337. MR 3583357Google Scholar
Croot, E. and Sisask, O., A probabilistic technique for finding almost-periods of convolutions, Geom. Funct. Anal. 20 (2010), no. 6, 1367–1396. MR 2738997CrossRefGoogle Scholar
Dong, D., Li, X., and Sawin, W., Improved estimates for polynomial Roth type theorems in finite fields, J. Anal. Math. 141 (2020), no. 2, 689–705. MR 4179774CrossRefGoogle Scholar
Ellenberg, J. S. and Gijswijt, D., On large subsets of Fnq with no three-term arithmetic progression, Ann. of Math. (2) 185 (2017), no. 1, 339–343. MR 3583358Google Scholar
Fox, J., A new proof of the graph removal lemma, Ann. of Math. (2) 174 (2011), no. 1, 561–579. MR 2811609CrossRefGoogle Scholar
Fox, J. and Lovász, L. M., A tight bound for Green’s arithmetic triangle removal lemma in vector spaces, Adv. Math. 321 (2017), 287–297. MR 3715712CrossRefGoogle Scholar
Fox, J., M. Lovász, L., and Sauermann, L., A polynomial bound for the arithmetic k-cycle removal lemma in vector spaces, J. Combin. Theory Ser. A 160 (2018), 186–201. MR 3846201CrossRefGoogle Scholar
Frankl, P., Graham, R. L., and V. Rödl, On subsets of abelian groups with no 3-term arithmetic progression, J. Combin. Theory Ser. A 45 (1987), no. 1, 157–161. MR 883900CrossRefGoogle Scholar
Frantzikinakis, N., Multiple ergodic averages for three polynomials and applications, Trans. Amer. Math. Soc. 360 (2008), no. 10, 5435–5475. MR 2415080CrossRefGoogle Scholar
Furstenberg, H., Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Analyse Math. 31 (1977), 204–256. MR 0498471CrossRefGoogle Scholar
Furstenberg, H. and Katznelson, Y., An ergodic Szemerédi theorem for commuting transformations, J. Analyse Math. 34 (1978), 275–291 (1979). MR 531279CrossRefGoogle Scholar
Gowers, W. T., A new proof of Szemerédi’s theorem for arithmetic progressions of length four, Geom. Funct. Anal. 8 (1998), no. 3, 529–551. MR 1631259CrossRefGoogle Scholar
Gowers, W. T., Arithmetic progressions in sparse sets, Current developments in mathematics, 2000, Int. Press, Somerville, MA, 2001, pp. 149–196. MR 1882535Google Scholar
Gowers, W. T., A new proof of Szemerédi’s theorem, Geom. Funct. Anal. 11 (2001), no. 3, 465–588. MR 1844079CrossRefGoogle Scholar
Gowers, W. T., Hypergraph regularity and the multidimensional Szemerédi theorem, Ann. of Math. (2) 166 (2007), no. 3, 897–946. MR 2373376CrossRefGoogle Scholar
Gowers, W. T., Green, B., Manners, F., and Tao, T., On a conjecture of marton, preprint (2023), arXiv:2311.05762.Google Scholar
Gowers, W. T. and Milićević, L., A quantitative inverse theorem for the U4 norm over finite fields, preprint (2017), arXiv:1712.00241.Google Scholar
Gowers, W. T. and Milićević, L., An inverse theorem for Freiman multi-homomorphisms, preprint (2020), arXiv:2002.11667.Google Scholar
Gowers, W. T. and Wolf, J., The true complexity of a system of linear equations, Proc. Lond. Math. Soc. (3) 100 (2010), no. 1, 155–176. MR 2578471CrossRefGoogle Scholar
Gowers, W. T. and Wolf, J., Linear forms and higher-degree uniformity for functions on Fnp, Geom. Funct. Anal. 21 (2011), no. 1, 36–69. MR 2773103CrossRefGoogle Scholar
Gowers, W. T. and Wolf, J., Linear forms and quadratic uniformity for functions on Fnq, Mathe-matika 57 (2011), no. 2, 215–237. MR 2825234Google Scholar
Gowers, W. T. and Wolf, J., Linear forms and quadratic uniformity for functions on ZN, J. Anal. Math. 115 (2011), 121–186. MR 2855036CrossRefGoogle Scholar
Green, B., Finite field models in additive combinatorics, Surveys in combinatorics 2005, London Math. Soc. Lecture Note Ser., vol. 327, Cambridge Univ. Press, Cambridge, 2005, pp. 1–27. MR 2187732Google Scholar
Green, B., Notes on the polynomial freiman–ruzsa conjecture, unpublished notes (2005), http://people.maths.ox.ac.uk/greenbj/papers/PFR.pdf.Google Scholar
Green, B., A Szemerédi-type regularity lemma in abelian groups, with applications, Geom. Funct. Anal. 15 (2005), no. 2, 340–376. MR 2153903CrossRefGoogle Scholar
Green, B., Montreál notes on quadratic Fourier analysis, Additive combinatorics, CRM Proc. Lecture Notes, vol. 43, Amer. Math. Soc., Providence, RI, 2007, pp. 69–102. MR 2359469Google Scholar
Green, B., Sárközy’s theorem in function fields, Q. J. Math. 68 (2017), no. 1, 237–242. MR 3658291CrossRefGoogle Scholar
Green, B., On Sárközy’s theorem for shifted primes, preprint (2022), arXiv:2206.08001.Google Scholar
Green, B., Manners, F., and Tao, T., Sumsets and entropy revisited, preprint (2023), arXiv:2306.13403.Google Scholar
Green, B. and Tao, T., An inverse theorem for the Gowers U 3(G) norm, Proc. Edinb. Math. Soc. (2) 51 (2008), no. 1, 73–153. MR 2391635CrossRefGoogle Scholar
Green, B. and Tao, T., The distribution of polynomials over finite fields, with applications to the Gowers norms, Contrib. Discrete Math. 4 (2009), no. 2, 1–36. MR 2592422Google Scholar
Green, B. and Tao, T., New bounds for Szemerédi’s theorem. I. Progressions of length 4 in finite field geometries, Proc. Lond. Math. Soc. (3) 98 (2009), no. 2, 365–392. MR 2481952CrossRefGoogle Scholar
Green, B. and Tao, T., An arithmetic regularity lemma, an associated counting lemma, and applications, An irregular mind, Bolyai Soc. Math. Stud., vol. 21, Jańos Bolyai Math. Soc., Budapest, 2010, pp. 261–334. MR 2815606CrossRefGoogle Scholar
Green, B. and Tao, T., An equivalence between inverse sumset theorems and inverse conjectures for the U3 norm, Math. Proc. Cambridge Philos. Soc. 149 (2010), no. 1, 1–19. MR 2651575CrossRefGoogle Scholar
Green, B. and Tao, T., New bounds for Szemerédi’s theorem, Ia: Progressions of length 4 in finite field geometries revisited, preprint (2012), arXiv:1205.1330.Google Scholar
Green, B., Tao, T., and Ziegler, T., An inverse theorem for the Gowers U4-norm, Glasg. Math. J. 53 (2011), no. 1, 1–50. MR 2747135CrossRefGoogle Scholar
Green, B., Tao, T., and Ziegler, T., An inverse theorem for the Gowers Us+1[N ]-norm, Ann. of Math. (2) 176 (2012), no. 2, 1231–1372. MR 2950773CrossRefGoogle Scholar
Han, R., Lacey, M. T., and Yang, F., A polynomial Roth theorem for corners in finite fields, Mathematika 67 (2021), no. 4, 885–896. MR 4304416CrossRefGoogle Scholar
Heath-Brown, D. R., Integer sets containing no arithmetic progressions, J. London Math. Soc. (2) 35 (1987), no. 3, 385–394. MR 889362CrossRefGoogle Scholar
Hosseini, K. and Lovett, S., A bilinear Bogolyubov-Ruzsa lemma with polylogarithmic bounds, Discrete Anal. (2019), Paper No. 10, 14. MR 3975362Google Scholar
Janzer, O., Low analytic rank implies low partition rank for tensors, preprint (2018), arXiv:1809.10931.Google Scholar
Janzer, O., Polynomial bound for the partition rank vs the analytic rank of tensors, Discrete Anal. (2020), Paper No. 7, 18. MR 4107323Google Scholar
Kaufman, T. and Lovett, S., Worst case to average case reductions for polynomials, 49th Annual IEEE symposium on Foundations of Computer Science, 2008, pp. 166–175.CrossRefGoogle Scholar
Kazhdan, D. and Ziegler, T., Approximate cohomology, Selecta Math. (N.S.) 24 (2018), no. 1, 499–509. MR 3769736CrossRefGoogle Scholar
Kelley, Z. and Meka, R., Strong bounds for 3-progressions, preprint (2023), arXiv:2302.05537.Google Scholar
Kim, D., Li, A., and Tidor, J., Cubic Goldreich-Levin, Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), SIAM, Philadelphia, PA, 2023, pp. 4846–4892. MR 4538136CrossRefGoogle Scholar
Kleinberg, R., Sawin, W., and D. E. Speyer, The growth of tri-colored sum-free sets, Discrete Anal. (2018), Paper No. 12, 10. MR 3827120Google Scholar
Král, D., Serra, O., and Vena, L., A combinatorial proof of the removal lemma for groups, J. Combin. Theory Ser. A 116 (2009), no. 4, 971–978. MR 2513645CrossRefGoogle Scholar
Kuca, B., Further bounds in the polynomial Szemerédi theorem over finite fields, Acta Arith. 198 (2021), no. 1, 77–108. MR 4214350CrossRefGoogle Scholar
Kuca, B., Multidimensional polynomial Szemerédi theorem in finite fields for polynomials of distinct degrees, preprint (2021), arXiv:2103.12606.Google Scholar
Kuca, B., True complexity of polynomial progressions in finite fields, Proc. Edinb. Math. Soc. (2) 64 (2021), no. 3, 448–500. MR 4330272CrossRefGoogle Scholar
Kuca, B., Multidimensional polynomial patterns over finite fields: bounds, counting estimates and Gowers norm control, preprint (2023), arXiv:2304.10793.Google Scholar
Kuca, B., On several notions of complexity of polynomial progressions, Ergodic Theory Dynam. Systems 43 (2023), no. 4, 1269–1323. MR 4555828CrossRefGoogle Scholar
Lacey, M. T. and W. McClain, On an argument of Shkredov on two-dimensional corners, Online J. Anal. Comb. (2007), no. 2, Art. 2, 21. MR 2289954Google Scholar
Lê, T. H. and Y.-R. Liu, On sets of polynomials whose difference set contains no squares, Acta Arith. 161 (2013), no. 2, 127–143. MR 3141915CrossRefGoogle Scholar
Leng, J., A quantitative bound for Szemerédi’s theorem for a complexity one polynomial progression over Z/N Z, preprint (2019), arXiv:1903.02592.Google Scholar
Li, A. and Sauermann, L., Sárközy’s theorem in various finite field settings, preprint (2022), arXiv:2212.12754.Google Scholar
Lovett, S., Equivalence of polynomial conjectures in additive combinatorics, Combinatorica 32 (2012), no. 5, 607–618. MR 3004811CrossRefGoogle Scholar
Lovett, S., An exposition of Sanders’ quasi-polynomial Freiman-Ruzsa theorem, Graduate Surveys, no. 6, Theory of Computing Library, 2015.Google Scholar
Lovett, S., The analytic rank of tensors and its applications, Discrete Anal. (2019), Paper No. 7, 10. MR 3964143Google Scholar
Lovett, S., Meshulam, R., and Samorodnitsky, A., Inverse conjecture for the Gowers norm is false, Theory Comput. 7 (2011), 131–145. MR 2862496CrossRefGoogle Scholar
Manners, F., Good bounds in certain systems of true complexity one, Discrete Anal. (2018), Paper No. 21, 40. MR 3900336Google Scholar
Manners, F., Quantitative bounds in the inverse theorem for the Gowers Us+1-norms over cyclic groups, preprint (2018), arXiv:1811.00718.Google Scholar
Manners, F., True complexity and iterated Cauchy–Schwarz, preprint (2021), arXiv:2109.05731.Google Scholar
Meshulam, R., On subsets of finite abelian groups with no 3-term arithmetic progressions, J. Combin. Theory Ser. A 71 (1995), no. 1, 168–172. MR 1335785CrossRefGoogle Scholar
Milićević, L., Polynomial bound for partition rank in terms of analytic rank, Geom. Funct. Anal. 29 (2019), no. 5, 1503–1530. MR 4025518CrossRefGoogle Scholar
Milićević, L., Quantitative inverse theorem for Gowers uniformity norms U52 and U6 in Fnq, preprint (2022), arXiv:2207.01591.Google Scholar
Moshkovitz, G. and Zhu, D. G., Quasi-linear relation between partition and analytic rank, preprint (2022), arXiv:2211.05780.Google Scholar
Nagle, B., Rödl, V., and Schacht, M., The counting lemma for regular k-uniform hypergraphs, Random Structures Algorithms 28 (2006), no. 2, 113–179. MR 2198495CrossRefGoogle Scholar
Naslund, E., Exponential bounds for the Erdős-Ginzburg-Ziv constant, J. Combin. Theory Ser. A 174 (2020), 105185, 19. MR 4078996CrossRefGoogle Scholar
Naslund, E., The partition rank of a tensor and k-right corners in Fnq, J. Combin. Theory Ser. A 174 (2020), 105190, 25. MR 4078997CrossRefGoogle Scholar
Norin, S., A distribution on triples with maximum entropy marginal, Forum Math. Sigma 7 (2019), e46, 12. MR 4061969CrossRefGoogle Scholar
Pebody, L., Proof of a conjecture of Kleinberg-Sawin-Speyer, Discrete Anal. (2018), Paper No. 13, 7. MR 3827119Google Scholar
Peluse, S., Three-term polynomial progressions in subsets of finite fields, Israel J. Math. 228 (2018), no. 1, 379–405. MR 3874848CrossRefGoogle Scholar
Peluse, S., On the polynomial Szemerédi theorem in finite fields, Duke Math. J. 168 (2019), no. 5, 749–774. MR 3934588CrossRefGoogle Scholar
Peluse, S., Bounds for sets with no polynomial progressions, Forum Math. Pi 8 (2020), e16, 55. MR 4199235Google Scholar
Peluse, S., Recent progress on bounds for sets with no three terms in arithmetic progression, Astérisque (2022), no. 438, No. 1196, 581. MR 4576028Google Scholar
Peluse, S., Subsets of Fpn × Fpn without L-shaped configurations, Compos. Math. 160 (2024), no. 1, 176–236.CrossRefGoogle Scholar
Peluse, S. and Prendiville, S., Quantitative bounds in the non-linear Roth theorem, preprint (2019), arXiv:1903.02592.Google Scholar
Peluse, S. and Prendiville, S., A polylogarithmic bound in the nonlinear Roth theorem, Int. Math. Res. Not. IMRN (2020), rnaa261.CrossRefGoogle Scholar
Prendiville, S., Quantitative bounds in the polynomial Szemerédi theorem: the homogeneous case, Discrete Anal. (2017), no. 5.CrossRefGoogle Scholar
Rödl, V. and Skokan, J., Regularity lemma for k-uniform hypergraphs, Random Structures Algorithms 25 (2004), no. 1, 1–42. MR 2069663CrossRefGoogle Scholar
Roth, K. F., On certain sets of integers, J. London Math. Soc. 28 (1953), 104–109. MR 0051853CrossRefGoogle Scholar
Ruzsa, I. Z., Difference sets without squares, Period. Math. Hungar. 15 (1984), no. 3, 205–209. MR 756185CrossRefGoogle Scholar
Ruzsa, I. Z., Sumsets and entropy, Random Structures & Algorithms 34 (2009), no. 1, 1–10.CrossRefGoogle Scholar
Samorodnitsky, A., Low-degree tests at large distances, STOC’07—Proceedings of the 39th Annual ACM Symposium on Theory of Computing, ACM, New York, 2007, pp. 506–515. MR 2402476CrossRefGoogle Scholar
Sanders, T., Additive structures in sumsets, Math. Proc. Cambridge Philos. Soc. 144 (2008), no. 2, 289–316. MR 2405891CrossRefGoogle Scholar
Sanders, T., On Roth’s theorem on progressions, Ann. of Math. (2) 174 (2011), no. 1, 619–636. MR 2811612CrossRefGoogle Scholar
Sanders, T., On certain other sets of integers, J. Anal. Math. 116 (2012), 53–82. MR 2892617CrossRefGoogle Scholar
Sanders, T., On the Bogolyubov-Ruzsa lemma, Anal. PDE 5 (2012), no. 3, 627–655. MR 2994508CrossRefGoogle Scholar
Sárközy, A., On difference sets of sequences of integers. III, Acta Math. Acad. Sci. Hungar. 31 (1978), 355–386.CrossRefGoogle Scholar
Schoen, T., Improved bound in Roth’s theorem on arithmetic progressions, Adv. Math. 386 (2021), Paper No. 107801, 20. MR 4266746CrossRefGoogle Scholar
Schoen, T. and Sisask, O., Roth’s theorem for four variables and additive structures in sums of sparse sets, Forum Math. Sigma 4 (2016), Paper No. e5, 28. MR 3482282CrossRefGoogle Scholar
Shkredov, I. D., On a generalization of Szemerédi’s theorem, Proc. London Math. Soc. (3) 93 (2006), no. 3, 723–760. MR 2266965CrossRefGoogle Scholar
Shkredov, I. D., On a problem of Gowers, Izv. Ross. Akad. Nauk Ser. Mat. 70 (2006), no. 2, 179–221. MR 2223244Google Scholar
Szemerédi, E., On sets of integers containing no k elements in arithmetic progression, Acta Arith. 27 (1975), 199–245, Collection of articles in memory of JuriĭVladimirovič Linnik. MR 0369312CrossRefGoogle Scholar
Szemerédi, E., Integer sets containing no arithmetic progressions, Acta Math. Hungar. 56 (1990), no. 1-2, 155–158. MR 1100788CrossRefGoogle Scholar
Tao, T., A variant of the hypergraph removal lemma, J. Combin. Theory Ser. A 113 (2006), no. 7, 1257–1280. MR 2259060CrossRefGoogle Scholar
Tao, T., Sumset and inverse sumset theory for Shannon entropy, Comb. Probab. Comput. 19 (2010), no. 4, 603–639.CrossRefGoogle Scholar
Tao, T., Higher order Fourier analysis, Graduate Studies in Mathematics, vol. 142, American Mathematical Society, Providence, RI, 2012. MR 2931680Google Scholar
Tao, T., A symmetric formulation of the Croot–Lev–Pach–Ellenberg–Gijswijt capset bound, https://terrytao.wordpress.com/2016/05/18/a-symmetricformulation-of-the-croot-lev-pach-ellenberg-gijswijt-capset-bound/, 2016, Accessed: 2021-05-17.Google Scholar
Tao, T. and Ziegler, T., The inverse conjecture for the Gowers norm over finite fields via the correspondence principle, Anal. PDE 3 (2010), no. 1, 1–20. MR 2663409CrossRefGoogle Scholar
Tao, T. and Ziegler, T., The inverse conjecture for the Gowers norm over finite fields in low characteristic, Ann. Comb. 16 (2012), no. 1, 121–188. MR 2948765CrossRefGoogle Scholar
Tidor, J., Quantitative bounds for the U4-inverse theorem over low characteristic finite fields, Discrete Anal. (2022), Paper No. 14, 17. MR 4503221Google Scholar
Tyrrell, F., New lower bounds for cap sets, preprint (2022), arXiv:2209.10045.Google Scholar
Wolf, J., Finite field models in arithmetic combinatorics—ten years on, Finite Fields Appl. 32 (2015), 233–274. MR 3293412CrossRefGoogle Scholar

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×