# Tom Sanders

Mathematician, Oxford.

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

I am a Senior Research Fellow of the Mathematical Institute at the University of Oxford, a University Research Fellow with the Royal Society, and a Tutorial Fellow of St Hugh's College, Oxford. My research interests lie in aspects of algebra, analysis, combinatorics, geometry and number theory, with particular emphasis on using tools from the former to address problems from the latter.

Anyone looking to contact me with regard to research can do so at tom.sanders@maths.ox.ac.uk or use the more traditional contact details available on my departmental page. Similarly, for college matters there is tom.sanders@st-hughs.ox.ac.uk and my college page.

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 and Zentralblatt MATH. 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 2011–2012, 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.

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

### Mathematical reviews

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

Mathematics written in LaTeX and displayed by MathJax; web-traffic data collected by Google Analytics; HTML validated as HTML 4.01 Transitional by the W3C Markup Validation Service; and CSS validated as CSS level 3 by the W3C CSS Validation Service.

Thanks to Julia Wolf for assistance in developing this site.