# Tom Sanders

Research mathematician, Oxford.

### Publications - Lecture courses - Professional duties - CV (pdf)

I am a research mathematician interested in aspects of algebra, analysis, combinatorics, geometry and number theory, with particular emphasis on using tools from the former to address problems in the latter. I completed my PhD in additive combinatorics under the direction of W. T. Gowers in 2007, during which time I benefited considerably from conversations and collaboration with B. J. Green.

Currently I am a senior research fellow in the Mathematical Institute at the University of Oxford, a university research fellow with the Royal Society, and a tutorial fellow at St. Hugh's College, Oxford. Between 2006 and 2011 I was a research fellow at Christ's College, Cambridge working in the Department of Pure Mathematics and Mathematical Statistics at the University of Cambridge. For the academic year 2007–2008 I was a member of the School of Mathematics at the Institute for Advanced Study, Princeton; for the fall semester 2008–2009 I was a member of the Mathematical Sciences Research Institute, Berkeley; and for the spring semester 2008–2009 I was a member of the Institut Mittag-Leffler, Stockholm.

Anyone interested in my work is welcome to contact me either through my departmental page, my Mendeley profile or by e-mail at tom.sanders@maths.ox.ac.uk.

The following list may also be found on arXiv in the order in which the papers were updated starting with the most recent update. Partial versions of the list appear on MathSciNet, Zentralblatt MATH, and my Mendeley profile. Note that some of the MathSciNet and Zentralblatt links require subscriptions to access in their entirety.

ArXiv versions of papers are at least as up-to-date as those linked to from DOIs, although journal formatting is not included.

 [1] The $\ell^1$-norm of the Fourier transform on compact vector spaces. Bull. Lond. Math. Soc. 39 (2007), no. 3, 509–521. arXiv:math/0605519. doi:10.1112/blms/bdm010. MR2331582. Zbl 1127.42001. [2] Additive structures in sumsets. Math. Proc. Camb. Philos. Soc. 144 (2008), no. 2, 289–316. arXiv:math/0605520. doi:10.1017/S030500410700093X. MR2405891. Zbl 05285594. [3] The Littlewood-Gowers problem. J. Anal. Math. 101 (2007), 123–162. arXiv:math/0605522. doi:10.1007/s11854-007-0005-1. MR2346542. Zbl 05243800. [4] An application of a local version of Chang's theorem. arXiv:math/0607668. Supplement to [1] using ideas in [2]. [5] A note on Freĭman's theorem in vector spaces. Combin. Probab. Comput. 17 (2008), no. 2, 297–305. arXiv:math/0605523. doi:10.1017/S0963548307008644. MR2396355. Zbl 1151.15003. [6] (with B. J. Green) Boolean functions with small spectral norm. Geom. Funct. Anal. 18 (2008), no. 1, 144–162. arXiv:math/0605524. doi:10.1007/s00039-008-0654-y. MR2399099. Zbl 05275311. [7] (with B. J. Green) A quantitative version of the idempotent theorem in harmonic analysis. Ann. of Math. (2) 168 (2008), no. 3, 1025–1054. arXiv:math/0611286. doi:10.4007/annals.2008.168.1025. MR2456890. Zbl 1170.43003. [8] Three-term arithmetic progressions and sumsets. Proc. Edinb. Math. Soc. (2) 52 (2009), no. 1, 211–233. arXiv:math/0611304. doi:10.1017/S0013091506001398. MR2475890. Zbl 05522642. [9] Appendix to `Roth's theorem on progressions revisited' by J. Bourgain. J. Anal. Math. 104 (2008), 193–206. arXiv:0710.0642. doi:10.1007/s11854-008-0021-9. MR2403434. Zbl 05320474. Appendix to [B]. [10] (with I. Z. Ruzsa) Difference sets and the primes. Acta Arith. 131 (2008), no. 3, 281–301. arXiv:0710.0644. doi:10.4064/aa131-3-5. MR2379933. Zbl 1170.11023. [11] A Freĭman-type theorem for locally compact abelian groups. Ann. Inst. Fourier (Grenoble) 59 (2009), no. 4, 1321–1335. arXiv:0710.2545. MR2566962. Zbl 1179.43002. [12] On a theorem of Shkredov. Online J. Anal. Comb. No. 5 (2010), Art. 6, 4 pp. arXiv:0807.5100. MR2789255. Remarks on the paper [Sh]. [13] Roth's theorem in $\mathbb{Z}_4^n$. Anal. PDE 2 (2009), no. 2, 211–234. arXiv:0807.5101. doi:10.2140/apde.2009.2.211. MR2560257. Zbl 05668750. [14] Chowla's cosine problem. Israel J. Math. 179 (2010), 1–28. arXiv:0807.5104. doi:10.1007/s11856-010-0071-4. MR2735033. Zbl 05823080. [15] Popular difference sets. Online J. Anal. Comb. No. 5 (2010), Art. 5, 4 pp. arXiv:0807.5106. MR2789268. Remarks on the paper [W]. [16] Approximate groups and doubling metrics. Math. Proc. Cambridge Philos. Soc. 152 (2012), no. 3, 385–404. arXiv:0912.0305. doi:10.1017/S0305004111000740. MR2911137. Zbl 06033590. [17] On a non-abelian Balog-Szemerédi-type lemma. J. Aust. Math. Soc. 89 (2010), no. 1, 127–132. arXiv:0912.0306. doi:10.1017/S1446788710000236. MR2727067. Zbl 05807816. Application of a key idea in [KK]; cf. [Lemma 3, S]. [18] A quantitative version of the non-abelian idempotent theorem. Geom. Funct. Anal. 21 (2011), 141–221. arXiv:0912.0308. doi:10.1007/s00039-010-0107-2. MR2773105. Zbl 1213.43007 . [19] Structure in sets with logarithmic doubling. Canad. Math. Bull., to appear. arXiv:1002.1552. [20] Green's sumset problem at density one half. Acta Arith. 146 (2011), no. 1, 91–101. arXiv:1003.5649. doi:10.4064/aa146-1-6. MR2741192. Zbl 05834870. [21] On certain other sets of integers. J. Anal. Math. 116 (2012), 53–82. arXiv:1007.5444. doi:10.1007/s11854-012-0003-9. MR2892617. [22] On Roth's theorem on progressions. Ann. of Math. (2) 174 (2011), no. 1, 619–636. arXiv:1011.0104. doi:10.4007/annals.2011.174.1.20. MR2811612. Zbl 059607140. [23] On the Bogolyubov-Ruzsa lemma. Anal. PDE 5 (2012), no. 3, 627–655. arXiv:1011.0107. doi:10.2140/apde.2012.5.627. Application of a key idea in [CS]. [24] Approximate (Abelian) groups. Proceedings of 6ECM, to appear. arXiv:1212.0456. [25] An analytic approach to a weak non-Abelian Kneser-type theorem. arXiv:1212.0457. [26] The structure theory of set addition revisited. Bull. Amer. Math. Soc. 50 (2013), 93–127. arXiv:1212.0458. doi:10.1090/S0273-0979-2012-01392-7.

 [B] J. Bourgain. Roth's theorem on progressions revisited. J. Anal. Math., 104 (2008), 155–192. doi:10.1007/s11854-008-0020-x. MR2403433. Zbl 1155.11011. [CS] E. S. Croot and O. Sisask. A probabilistic technique for finding almost-periods of convolutions. Geom. Funct. Anal. 20 (2010), 1367–1396. arXiv:1003.2978. doi:10.1007/s00039-010-0101-8. MR2738997. Zbl 05833796. [KK] N. H. Katz and P. Koester. On additive doubling and energy. SIAM J. Discrete Math., 24(4):1684–1693. arXiv:0802.4371. doi:10.1137/080717286. MR2746716. Zbl 05924702. [S] T. Schoen. Near optimal bounds in Freĭman's theorem. Duke Math. J. 158 (2011), no. 1, 1–12. doi:10.1215/00127094-1276283. MR2794366. Zbl 05904309. [Sh] I. D. Shkredov. On sets with small doubling. (Russian. Russian summary) Mat. Zametki 84 (2008), no. 6, 927–947; translation in Math. Notes 84 (2008), no. 5-6, 859–878. arXiv:math/0703309. doi:10.1134/S000143460811028X. MR2492806. Zbl 1219.11019. [W] J. Wolf. The structure of popular difference sets. Israel J. Math. 179 (2010), 253–278. doi:10.1007/s11856-010-0081-2. MR2735043. Zbl 05823090.

Courses marked as [ongoing] will have their notes etc. updated as they proceed.

Topics in analytic number theory. Part III, Lent 2009–2010, University of Cambridge. notes; examples sheet; exam.
Lectures: Wednesdays, 13.00–14.00, MR5; Thursdays, 11.00–12.00, MR4. Examples class(es): 28th April 2010, 14.00–16.00, MR14, CMS.
Analysis of Boolean functions. Part III, Michaelmas 2010–2011, University of Cambridge. notes; examples sheet; exam.
Lectures: Tuesdays and Thursdays, 12.00–13.00, MR13, CMS. Examples class(es): 2nd December 2010, 14.00–15.00, MR13, CMS; 11th May 2011, 14.00–16.00, MR5, CMS.
Applications of commutative harmonic analysis. MFoCS, Trinity 2012–2011, University of Oxford. notes; examples sheet; exam.
Lectures: Tuesdays and Thursdays, 11.00–12.00, L3, Mathematical Institute. Examples class(es): 15th May 2012, 13.00–14.00, L3, Mathematical Institute; 5th June 2012, 13.15–14.15, L3, Mathematical Institute.

This section covers editorial advice, invited talks, mathematical reviews and seminar series organisation.

Moscow Journal of Combinatorics and Number Theory. Editorial board member, 2010–Present. publication topics.
London Mathematical Society. Editorial advisor, 2013–Present.

### Invited talks

Analysis seminar, University of Cambridge, 9th November 2005.
Pure mathematics seminar, University College London, 11th January 2006.
Combinatorics seminar, University of Cambridge, 26th January 2006.
Additive combinatorics workshop, Centre de Recherches Mathématiques, Université de Montréal, 11th April 2006.
Pure mathematics seminar, Royal Holloway, University of London, 9th May 2006.
Harmonic analysis and PDEs workshop, University of Birmingham, 12th September 2007.
Arithmetic combinatorics workshop, Institute for Advanced Study, Princeton, 2nd October 2007.
Theory of computer science seminar, Courant Institute of Mathematical Sciences, New York, 15th November 2007.
Number theory seminar, Stanford University, 22nd February 2008.
Additive combinatorics, number theory, and harmonic analysis workshop, Fields Institute, Toronto, 11th April 2008.
Number theory seminar, City University of New York, 1st May 2008.
Dynamical systems seminar, Northwestern University, 20th May 2008.
Introduction to ergodic theory and additive combinatorics workshop, Mathematical Sciences Research Institute, Berkeley, 28th August 2008.
Discrete rigidity phenomena in additive combinatorics workshop, Mathematical Sciences Research Institute, Berkeley, 4th November 2008.
Analysis and PDE seminar, University of California, Los Angeles, 12th December 2008.
Discrete probability seminar, Institut Mittag-Leffler, 20th January 2009.
Dynamical systems, Number theory and Analysis seminar, Kungliga Tekniska högskolan, 2nd July 2009.
Analytical methods in combinatorics, additive number theory and computer science, Institute for Pure and Applied Mathematics, 2nd December 2009.
Number theory seminar, University of Oxford, 11th March 2010.
Probability and statistics seminar, Uppsala Universitet, 15th April 2010.
Combinatorics seminar, University of Oxford, 1st June 2010.
Discrete analysis seminar, University of Cambridge, 23rd November 2010.
Séminaire CAESAR de combinatoire additive, École Polytechnique, 10th February 2011.
Heilbronn seminar, University of Bristol, 2nd March 2011.
Discrete harmonic analysis workshop, Isaac Newton Institute for Mathematical Sciences, 30th March 2011.
One day colloquium in combinatorics, London School of Economics, 19th May 2011.
Young workshop in arithmetics and combinatorics, Instituto de Ciencias Matemáticas, Madrid, 22nd June 2011.
Groups and additive combinatorics satellite workshop, Gregynog Hall, University of Wales, 28th June 2011.
Paul Turán memorial conference, Alfréd Rényi Institute of Mathematics, Budapest, 24th August 2011.
Séminaire de théorie des nombres, Institut Élie Cartan, Nancy, 17th November 2011.
Combinatorics seminar, University of Oxford, 22nd November 2011.
Combinatorics seminar, University of Birmingham, 26th January 2012.
Discrete analysis seminar, University of Cambridge, 17th February 2012.
European Congress of Mathematics, Kraków, 4th July 2012.
Combinatoire Additive, Institut Henri Poincaré, 9th July 2012.
Oxford number theory day, University of Oxford, 13th October 2012.
Mathematics colloquium, University of Warwick, 26th October 2012.

### Mathematical reviews

MathSciNet. Reviewer, 2006–Present. reviews.
Zentralblatt MATH. Reviewer, 2008–2010. reviews.

### Seminar series

Discrete Analysis seminar. 2006/2007, University of Cambridge. Co-organiser(s): W. T. Gowers and B. J. Green.
Analysis seminar. 2007/2008, Institute for Advanced Study, Princeton. Co-organiser(s): J. Wolf.
Post-doctoral seminar. 2008/2009, Mathematical Sciences Research Institute, Berkeley.
Discrete Analysis seminar. 2010/2011, University of Cambridge. Co-organiser(s): B. Bukh, P. Candela Pokorna and B. J. Green.

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Thanks to Julia Wolf for assistance in developing this site.