## Publications

### Book

B1. The book*The ternary Goldbach problem*(currently 561pp.+xvii) is to appear in

*Ann. of Math. Studies*. It will supersede preprints 26, 25 and 21 below. The current version differs substantially from the draft from 2015.

### Pure mathematics - research papers

41. Explicit bounds on sums of μ (avec A. Chirre), in preparation.40. New dimensional estimates for subvarieties of linear algebraic groups (avec J. Bajpai et D. Dona), Vietnam J. Math. 52, No. 2, 479-518 (2024).

39. Expansion, divisibility and parity: an explanation,

*Combinatorial and Additive Number Theory V*, Springer, Springer Proc. Math. Stat. 395, 199-237 (2022).

38. Growth estimates and diameter bounds for classical Chevalley groups (with J. Bajpai et D. Dona), submitted.

37. Expansion, divisibility and parity (with M. Radziwiłł), submitted.

36. Summing μ(n): a faster elementary algorithm (with L. Thompson),

*Res. Number Theory*9, no 1, paper no. 6 (2023), 37 p.

35. Optimality for the two-parameter quadratic sieve (with E. Carneiro, A. Chirre and J. Mejia-Cordero),

*Acta Arith.*203 (2022), 195-226.

34. Explicit L² bounds for the Riemann ζ function (with D. Dona and S. Zúñiga Alterman),

*J. Théor. Nombres Bordeaux*34 (2022), 91-133.

33.

*The ternary Goldbach problem*, to appear in

*Ann. of Math. Studies*. See Book above.

32. An improved sieve of Eratosthenes,

*Math. Comput.*89 (2020), no. 321, 333-350.

31. Growth in linear algebraic groups and permutation groups: towards a unified perspective, in:

*Groups St. Andrews 2017 in Birmingham. Selected papers of the conference, Birmingham, UK, August 5-13, 2017*, Cambridge University Press, 2019.

30. Isomorphismes de graphes en temps quasi-polynomial (d’après Babai et Luks, Weisfeiler-Leman,…),

*Astérisque*407 (2019), Séminaire Bourbaki 2016/2017, 135–182. As is usual for a Bourbaki seminar, this paper is largely expository; what is new is simply a precise value for the exponent (namely, 3). A translation by J. Bajpai and D. Dona (with solutions and some supplementary exercises) is available.

29. (with K. Juschenko) Soficity, short cycles and the Higman group,

*Trans. Am. Math. Soc.*, 371.4 (2019): 2771-2795.

28. (with P. Candela) On the dimension of additive sets,

*Acta Arith.*167 (2015), no. 1, 91-100.

27. Random generators of the symmetric group: diameter, mixing time and spectral gap (with Á. Seress and A. Zuk),

*J. of Algebra.*421 (2015), 349-368.

26. The ternary Goldbach conjecture is true, preprint.

25. Major arcs for Goldbach's problem, preprint.

24. (with D. Platt) Numerical verification of the ternary Goldbach conjecture up to 8.875e30,

*Experiment. Math.*22 (2013), no. 4, 406-409.

23. Growth in groups: ideas and perspectives,

*Bull. Am. Math. Soc.*52 (2015), no. 3, 357-413.

22. (with N. Gill and M. Rudnev) On growth in an abstract plane,

*Proc. Am. Math. Soc.*143 (2015), no. 8, 3593-3602

21. Minor arcs for Goldbach's problem, preprint.

20. Bounds on the diameter of Cayley graphs of the symmetric group (with J. Bamberg, N. Gill, T. Hayes, Á. Seress and P. Spiga),

*J. Algebraic Combin.*

19. Square-free values of f(p), f cubic,

*Acta Math.*213 (2014), no. 1, 107-135.

Th3. Groupes, courbes et croissance, habilitation thesis, Paris-Sud (Orsay).

18. On the diameter of permutation groups (with Á. Seress),

*Annals of Math.*179 (2014), no. 2, 611-658.

17. Deterministic methods to find primes (as

*D. H. J. Polymath*, with T. Tao and E. Croot),

*Math. Comp.*81 (2012), no. 278, 1233-1246.

16. Growth in solvable subgroups of GL

_{r}(Z/pZ) (with N. Gill),

*Math. Annalen*360 (2014), no. 1-2, 157-208.

15. Growth of small generating sets in SL

_{n}(Z/pZ) (with N. Gill),

*Int. Math. Res. Notices.*, Vol. 2011, 4226--4251.

14. An explicit incidence theorem in F

_{p}(with M. Rudnev),

*Mathematika*, 57 (2011), no. 1, 135--145.

13. Improving Roth's theorem in the primes (with A. de Roton), Int. Math. Res. Notices. (2011), Vol. 2011, 767--783.

12. Growth in SL

_{3}(Z/pZ), J. Eur. Math. Soc. (JEMS), vol. 13, no. 3, pp. 761--851.

11. Power-free values, repulsion between points, differing beliefs and the existence of error, CRM Proceedings and Lecture Notes, v. 46 (2008), 81-88.

10. How small must ill-distributed sets be? (with A. Venkatesh), in: Chen, W. W. L. (ed.) et al.,

*Analytic number theory. Essays in honour of Klaus Roth on the occasion of his 80th birthday.*Cambridge University Press, 2009, pp. 224-234.

9. Growth and generation in SL

_{2}(Z/pZ),

*Ann. of Math.*167 (2008), 601-623.

8. The parity problem for irreducible polynomials, submitted.

7. Power-free values, large deviations and integer points on irrational curves,

*J. Théor. Nombres Bordeaux*19 (2007), 433-472.

6. The parity problem for reducible polynomials,

*J. London Math. Soc. (2)*73 (2006), 415-435.

5. Integral points on elliptic curves and 3-torsion in class groups (with A. Venkatesh),

*J. Amer. Math. Soc.*19 (2006), 527--550.

4. Root numbers and ranks over global function fields (with B. Conrad and K. Conrad),

*Adv. Math.*198 (2005), 684--731.

3. On the behaviour of root numbers in families of elliptic curves, submitted.

2. On the square-free sieve,

*Acta Arith.*115 (2004), 349-402.

Th2. Root numbers and the parity problem, Ph.D. thesis, Princeton University, April 2003, math.NT/0305435.

1. Enumeration of tilings of diamonds and hexagons with defects (with I. M. Gessel),

*Electron. J. Combin.*6 (1999), no. 1, R16.

Th1. Edge effects on local statistics in lattice dimers: a study of the Aztec diamond (finite case), senior thesis, Brandeis University, May 1998, math.CO/0007136.

### Pure mathematics - popularization and exposition

PE8. Expansión, divisibilidad y paridad: un paseo por los números y grafos (with A. Ubis), in preparation.PE7. Primos, paridad y análisis (with A. Ubis), to appear in

*Actas de la escuela AGRA III.*

PE6. Crecimiento y expansión en SL₂, in: M. Belolipetsky, H. A. Helfgott and C. G. Moreira (eds.),

*Actas de la escuela AGRA II: Aritmética, grupos y análisis*, Publicações Matemáticas no. 42, IMPA, 2022.

PE5. Growth and expansion in algebraic groups over finite fields, in:

*Analytic methods in arithmetic geometry: Arizona Winter School 2016,*Contemporary Mathematics, v. 740, AMS, 2019.

PE4. The ternary Goldbach problem,

*Proceedings of the International Congress of Mathematicians - Seoul 2014*, Vol. II, 391-418, Kyung Moon Sa, Seoul, 2014.

PE3. La conjecture de Goldbach ternaire. Translated by M. Bilu, revised by the author.

*Gaz. Math.*140 (2014), 5-18.

PE2. La conjetura débil de Goldbach,

*Gac. R. Soc. Mat. Esp.*16 (2013), no. 4.

PE1. Azar y aritmética,

*Monografías del Instituto de Matemática y Ciencias Afines*, v. 50, IMCA, Lima, Perú.

*Note: PE2, PE3 and (in part) PE4 are based on my exposition in English on the subject.*

See also Machine-Assisted Proofs, a report of a ICM 2018 panel discussion (section 4 is a short overview I wrote).

### Pedagogy and history

HP3. A modern vision of the work of Cardano and Ferrari on quartics (with M. Helfgott),*CONVERGENCE*, an online journal of the Mathematical Association of America, July 2009.

HP2. A noncalculus proof that Fermat's principle of least time implies the law of refraction (with M. Helfgott),

*Am. J. Phys.*70 (2002), no. 12, 1224-1225.

HP1. Maxima and minima before Calculus (with M. Helfgott),

*Pro Mathematica*XII (1998), nos. 23-24, 135-158.

### Data compression and information theory

*I am no longer active in these fields. My work on algorithmic number theory and graph theory can be found under the heading of pure mathematics above.*

I3. Lossless image compression by block matching (with J. A. Storer), Comput. J. 40 (1997), no. 2/3, 137-145.

I2. Asymmetry in Ziv/Lempel ’78 Parsing (with M. Cohn), 320-328, in: Compression and complexity of sequences: proceedings, 1997, IEEE, Los Alamitos, CA, IEEE Computer Society Press, 1997.

I1. On Maximal Parsings of Strings (with M. Cohn), 291-299, in: Proceedings DCC ’1997: Data Compression Conference, IEEE, Los Alamitos, CA, 1997.