Topics in Diophantine Approximation

18.784: Seminar in Number Theory, Spring 2026

Welcome to 18.784: Seminar in Number Theory! This course is a Communication Intensive in Mathematics (CI-M) for undergraduates at MIT.

This semester, we will explore Diophantine approximation — an area of mathematics that studies how close rational numbers can be to irrational numbers. The subject has very classical origins in number theory and has intersections with a breadth of mathematical disciplines including:

Algebraic geometry
Algebraic number theory
Analytic number theory
Complex analysis
p-adic analysis
Dynamical systems
Game theory

Prerequisites for the course:

Algebra 1 (18.701) or
Modern Algebra (18.703) and Linear Algebra (18.06 or 18.700).

Additional mathematical background in any of the following will be helpful but not required:

Complex analysis (18.04 or 18.112)
Real analysis (18.100)
Galois theory (18.702)
Elementary number theory (18.781)

Students will learn how to communicate specific topics in Diophantine approximation through blackboard talks, a short expository term paper, and peer review.

Course files (to be updated during the semester):

Google forms:

Background questionnaire
Presentation feedback forms and sign-up sheets are posted in the course Slack

The class meets in Room 2-151 from 2:30pm—4:00pm (ET) on Tuesdays and Thursdays.
Robin's office hours are in 2-238 from 3:00pm—4:00pm (ET) on Wednesdays.

To contact Robin, please send an email or a message in the course Slack.
The WRAP (communications) instructor, Emily Robinson, can be reached at erobin73@mit.edu.

Course schedule (subject to change)

Date	Speaker	Title	References
February 3 (Background questionnaire due 2:30pm on February 4)	Robin Zhang	Organizational meeting
	Emily Robinson	Communications workshop: preparing for presentations	[Ruf19]
	Robin Zhang	What is Diophantine approximation	[Bak22]§1 [Bur00]§1 [Sch80]§1
February 5	All students	3-minute introductory talks (favorite definition / example / theorem)
February 10	Ana Illanes Martinez De La Vega & Alicia Lin	What is a real number	[HW71]§4.1-4.3 [Rud76]§1 [Sch14]§1.1-1.3 [Sut15]§1.1-1.3
	Cheuk Hei Chu & Paul Gutkovich	What is a Farey sequence	[Dum22]§6.2.1 [HW71]§3.1-3.8
	Reina Wang & Tony Wu	What is modular arithmetic	[HW71]§5.2-5.5, 6.1 [IR90]§3-4, 7 [Ros11]§4-5, 9 [Smi23]§II
February 12	Fiona Lee & Julianna Lian	What is the pigeonhole principle	[Bic20] [Goe15]
	Diego Andrés Rivera Orona & Alex Yang	What is the Euclidean algorithm	[Ros11]§3.4
	Yina Wang & Zachary West	What is equidistribution	[KN74]§1.1
February 17	President's Day Tuesday (no meeting)	President's Day Tuesday (no meeting)
February 19	Benjamin Li & Adelmo Morrison Orozco	What is an algebraic integer	[Mur15] [Ogg10]§1
	Tiago Oliveira Marques & Ghaura Mahabaduge	What is a p-adic number	[Con06] [Gou97]§1-3
	Emily Robinson	Communications workshop: effective mathematics presentations	[Ben01]§1
February 24	Snow Day (no meeting)	Snow Day (no meeting)
February 26	Robin Zhang	Summary of Diophantine approximation through density and Dirichlet's theorem
	Fiona Lee	Continued fractions and convergents	[Bug12]§Appendix D [Cas57]§I.2-I.4 [Ros11]§12.2-12.4 [Sch80]§I.5
	Robin Zhang	Introduce paper assignment	[Ser03]
March 3	Paul Gutkovich	Lattices & Minkowski's theorem	[Cas57]§Appendix B [HW71]§3.9-3.11 [Kle10]§11 [Sch80]§IV.1
	Ana Illanes Martinez De La Vega	Approximations of algebraic numbers & Liouville numbers	[Bak22]§1.1 [Bug12]§Appendix E.1 [Gar13] [HW71]§11.6-11.7 [Kle10]§3.4 [Sch80]§V.1
March 5	Alex Yang	Simultaneous approximation of irrational numbers	[Cas57]§I.5 [HW71]§11.12 [Sch80]§II
	Zachary West	Pell's equation & quadratic units	[Lem21]§2.3, 7.1-7.2 [IR90]§17.5 [Ros11]§13.4
	Emily Robinson	Communications reading workshop: finding sources for research papers	[Rob26a]
March 10	Yina Wang	p-adic approximation	[Mah61]§IV.6-IV.9 [Rom24]§1-6
	Cheuk Hei Chu	Beatty sequences	[Gar77] [Lee20]§4
	All students	Pre-workshop reading	[Bur00] [Cas57]§1
March 12	Ghaura Mahabaduge	Hurwitz's theorem & Markov's constant	[Ber14]§1.4 [Kle10]§3.2 [Sch80]§I.2, I.6
	Emily Robinson	Communications reading workshop: genre sources for research papers	[Bur00] [Cas57]§1 [PPTWZ09] [Rob26b] [Ruf08]
March 17	Alicia Lin	Badly approximable numbers & Schmidt's game	[Ber14]§1.5 [Dom20] [Kle10]§3.2, 9 [Sch80]§I.5, III.1-III.2
	Julianna Lian	The three-gap theorem	[PT03] [MS17] [van87] [van88] [WDMS25]
March 19 (Paper topic proposal due 2:30pm)	Reina Wang	Markov numbers	[Cas57]§II [Pro22]
	Tiago Oliveira Marques	LLL lattice reduction	[Kel09] [LLL82] [NV10]§1, 6, 10
March 24	Spring Break (no meeting)	Spring Break (no meeting)
March 26	Spring Break (no meeting)	Spring Break (no meeting)
March 31	Tony Wu	Khinchin's theorem & the metric viewpoint	[BvPSZ14]§3 [Cas57]§VII [Kle10]§4
	Adelmo Morrison Orozco	Digits of algebraic and transcendental numbers	[BMN70] [Bug12]§4, 8.0 [HW71]§IX [NS13]
April 2	Benjamin Li	Billiards and Diophantine approximation	[CO25]§1 [Gel09] [Put] [Sch23]
	Diego Andrés Rivera Orona	The Thue–Siegel–Roth theorem	[Cas57]§VI [HW71]§V.2-V.3 [IR90]§17.12
April 7 (Paper first draft due 2:30pm)	Tiago Oliveira Marques	The Coppersmith method: a reverse proof and applications	—
	Emily Robinson	Communications writing workshop: guiding text	—
April 9	Julianna Lian	Geometry of quasicrystals and Diophantine approximation	—
	Benjamin Li	Approximating reals by algebraic numbers	—
April 14	Cheuk Hei Chu	The lonely runner conjecture: connections to Diophantine approximation, geometry, and combinatorics in small cases	—
	Alex Yang	Breaking RSA with Diophantine approximation	—
April 16	Adelmo Morrison Orozco	Finding integer relations via PSLQ	—
	Emily Robinson	Communications writing workshop: writing economy	—
April 21	Alicia Lin	Cantor-winning sets	—
	Ghaura Mahabaduge	The lonely runner conjecture: reductions and computational challenges	—
April 23 (Paper second draft due 2:30pm)	Reina Wang	The discrete beginnings of the Lagrange spectrum and its connections to Markov theory	—
	Diego Andrés Rivera Orona	Feynman integrals and the PSLQ algorithm	—
April 28	Fiona Lee	The Gauss map and the distribution of partial quotients	—
	Paul Gutkovich	Ergodic theory and continued fractions	—
April 30 (Peer review reports due 2:30pm)	All students	Peer review on papers
May 5	Tony Wu	Geodesics on the modular surface and continued fractions	—
	Yina Wang	Intrinsic vs. extrinsic approximation of the middle-third Cantor set	—
May 7	Zachary West	Oppenheim's conjecture	—
	Ana Illanes Martinez De La Vega	The lonely runner conjecture: different approaches to proofs	—
May 12 (Paper final draft due 2:30pm)	All students	Course retrospective & celebration

Books

[Bak22] Alan Baker. Transcendental Number Theory. Cambridge University Press, 2022.
[Bic20] Allan Bickle. The Pigeonhole Principle. Supplement to Fundamentals of Graph Theory. Pure and Applied Undergraduate Texts, 43. American Mathematical Society, 2020.
[Bug12] Yann Bugeaud. Distribution Modulo One and Diophantine Approximation. Cambridge Tracts in Mathematics, 193. Cambridge University Press, 2012.
[Bur00] Edward B. Burger. Exploring the Number Jungle: A Journey into Diophantine Analysis. Student Mathematical Library, 8. American Mathematical Society, 2000.
[BvPSZ14] Jonathan Borwein, Alf van der Poorten, Jeffrey Shallit, and Wadim Zudilin. Neverending Fractions: An Introduction to Continued Fractions. Cambridge University Press, 2014
[Cas57] J.W.S. Cassels. An Introduction to Diophantine Approximation. Cambridge Tracts in Mathematics and Mathematical Physics, 45. Cambridge University Press, 1957.
[Gou97] Fernando Gouvêa. p-adic numbers. Second Edition. Universitext. Springer-Verlag, 1997.
[HW71] G. H. Hardy and E. M. Wright. An Introduction to the Theory of Numbers. Oxford University Press, 1971.
[IR90] Kenneth Ireland and Michael Rosen. A Classical Introduction to Modern Number Theory. Second Edition. Springer-Verlag, 1990.
[KN74] Lauwerens Kuipers and Harald Niederreiter. Uniform Distribution of Sequences. John Wiley & Sons, 1974.
[Lem21] Franz Lemmermeyer. Quadratic Number Fields. Springer Undergraduate Mathematics Series. Springer-Verlag, 2021.
[Mah61] Kurt Mahler. Lectures on Diophantine Approximations. Notre Dame Mathematical Lectures, 7. University of Notre Dame, 1961.
[NV10] Phong Q Nguyen and Brigitte Vallée. The LLL Algorithm. Springer-Verlag, 2010.
[Ros11] Kenneth H. Rosen. Elementary Number Theory and Its Applications. Sixth Edition. Addison–Wesley, 2011.
[Rud76] Walter Rudin. Principles of Mathematical Analysis. McGraw-Hill, 1976.
[Sch80] Wolfgang M. Schmidt. Diophantine Approximation. Lecture Notes in Mathematics, 785. Springer-Verlag, 1980.
[van87] Tony van Ravenstein. Number sequences and phyllotaxis. PhD Thesis, University of Wollongong, 1987.

Articles

[BMN70] W. A. Beyer, N. Metropolis, and J. R. Neergaard. Statistical study of digits of some square roots of integers in various bases. Math. Comp., 24(110): 455–473, 1970.
[Bur00] Edward B. Burger. Diophantine Olympics and world champions: polynomials and primes down under. Amer. Math. Monthly, 107(9):822–829, 2000.
[CO25] Hongjia H. Chen and Hinke M. Osinga. Classification of periodic orbits for square and rectangular billiars. ANZIAM J., 67:e33, 2025.
[Gal03] Gregory Galperin. Playing pool with π (the number π from a billiard point of view). Regul. Chaotic Dyn., 8(4):375–394, 2003.
[Gar77] Martin Gardner. Mathematical Games. Sci. Am., 236(3):134–141, 1977.
[LLL82] Arjen Klaas Lenstra, Hendrik Willem Lenstra Jr., and László Lovász. Factoring polynomials with rational coefficients. Math. Ann., 261(3):515–534, 1982.
[MS17] Jens Marklof and Andreas Strömbergsson. The three gap theorem and the space of lattices. Amer. Math. Monthly, 124(8):741–745, 2017.
[NS13] Johan Sejr Brinch Nielsen and Jakob Grue Simonsen. An experimental investigation of the normality of irrational algebraic numbers. Math. Comp., 82(283): 1837–1858, 2013.
[PT03] Edita Pelantová and Reidun Twarock. Tiles in quasicrystals with cubic irrationality. J. Phys. A, 36(14):4091, 2003.
[Rom24] Giuliano Romeo. Continued fractions in the field of p-adic numbers. Bull. Amer. Math. Soc., 61:343–371, 2024.
[Sch23] Richard Evan Schwartz. Survey lecture on billiards. ICM–International Congress of Mathematicians. Vol. 4. Sections 5–8, 2392–2429, 2023.
[van88] Tony van Ravenstein. The three-gap theorem (Steinhaus conjecture). J. Aust. Math. Soc., 45(3):360–370, 1988.
[WDMS25] Yinglai Wang, Bryn Davies, Marc Martí-Sabaté. Edge modes in discrete lattices: phase transitions under the three-gap theorem. Proc. R. Soc. A, 481(2318):20250019, 2025.

Notes

[Ber14] Victor Beresnevich. Metric number theory. University of York, 2014.
[Con06] Keith Conrad. The p-adic expansion of rational numbers. University of Connecticut, 2006.
[Dom20] Charlotte Dombrowsky. Schmidt's game applied to the set of badly approximable numbers. ETH Zurich, 2020.
[Dum22] Evan Dummit. Rational approximation and Diophantine equations. Northeastern University, 2022.
[Gar13] Paul Garrett. Liouville's theorem on Diophantine approximation. University of Minnesota, Twin Cities, 2013.
[Goe15] Michel Goemans. Pigeonhole principle and the probabilistic method. Massachusetts Institute of Technology, 2015.
[Kel09] Jonathan Kelner. An algorithmist's toolkit. Massachusetts Institute of Technology, 2009.
[Kle10] Dmitry Kleinbock. Metric Diophantine approximation and dynamical systems. Brandeis University, 2010.
[Lee20] Antony Lee. Diophantine Approximation and Dynamical Systems. Lund University, 2020.
[Mur15] Timothy Murphy. Algebraic numbers and algebraic integers. Trinity College Dublin, 2015.
[Ogg10] Frédérique Oggier. Introduction to algebraic number theory. Nanyang Technological University, 2010.
[Pro22] Yuri Prokhorov. Markov numbers in arithmetic and geometry. Steklov Mathematical Institute, 2022.
[Put] Andrew Putman. Periodic billiards paths on smooth tables. University of Notre Dame.
[Sch14] Richard Evan Schwartz. Dedekind cuts. Brown University, 2014.
[Smi23] Tim Smits. Introduction to Number Theory. University of California, Los Angeles, 2023.
[Sut15] Andrew Sutherland. p-adic numbers. Massachusetts Institute of Technology, 2015.

Communications resources

[Ben01] Robert L. Benedetto. An elementary product identity in polynomial dynamics. Amer. Math. Monthly, 108(9):860–864, 2001.
[Hom22] Jennifer Hom. Getting a handle on the Conway knot. Bull. Amer. Math. Soc. (N.S.), 59(1):19–29, 2022.
[PPTWZ09] Mike Paterson, Yuval Peres, Mikkel Thorup, Peter Winkler, and Uri Zwick. Maximum overhang (annotated). Amer. Math. Monthly, 116(9):763–787, 2009.
[Ser03] Jean-Pierre Serre. How to write mathematics badly (56 minutes). Harvard University, 2003. Transcript.
[Rob26a] Emily Robinson. Finding sources for research papers. Massachusetts Institute of Technology, 2026.
[Rob26b] Emily Robinson. Genre sources for research papers. Massachusetts Institute of Technology, 2026.
[Ruf08] Susan Ruff. Acknowledging sources in mathematics papers. Massachusetts Institute of Technology, 2008.
[Ruf19] Susan Ruff. Preparing a seminar chalk talk. Massachusetts Institute of Technology, 2019.