Part III Diophantine Analysis, Michaelmas 2024
This page contains information for students taking my lectures on Diophantine Analysis in Michaelmas 2024. The best way to contact me outside lectures is by email: pv270@dpmms.cam.ac.uk. (I do not monitor Moodle.)
What is this course about? A basic, but surprisingly useful fact is the following. Let \(a\neq b\in\mathbf{Z}\) be two distinct rational integers. Then their distance is at least \(1\): \(|a-b|\ge 1\). Can we give a better bound if we have more information about \( a\) and \( b\) ? What if we consider algebraic numbers instead of rational integers? We will discuss tools to answer such questions.
The course will be an introduction to the subspace theorem, linear forms in logarithms and their applications. In the first five lectures, I will introduce and state the main theorems and give some basic applications.
The next part of the course will give some insight into the proofs. I will prove some precursors of the main theorems, which already include many of the key ideas. I will prove Dyson's result on the irrationality exponent of algebraic numbers (but not via Dyson's lemma), and the Gelfond-Schneider theorem on the transcendence of numbers like \(\sqrt{2}^\sqrt{3}\) .
In the final part of the course, I will give more applications to various problems in number theory.
Prerequisites: Some knowledge of Galois theory, number fields and complex analysis will be assumed. If you are from Cambridge and took Part II Number Fields, then you probably also know enough Galois theory for this course. I wrote up a short summary of the definitions and facts about number fields that we use in the course. Chapters 10-12 in Baker's book offers a quick introduction to number fields, and covers more than what is needed for this course. A more thorough text is Marcus's book. My notes from last year are also available. Understanding the maximum modulus principle is the main prerequisite from complex analysis.
Lecture notes are available here. (Last updated: 20 December 2024.) The notes will be updated in the course of the term. Please check back, and see the changelog below. Non-examinable material is now clearly marked as such in the notes. It is explained at the beginning of the notes what this designation means.
Example sheets: The example sheets contain exercises to complement the course material. Solutions will be discussed at the example classes. The timetable will be published later (soon). You may submit your work for two questions that are designated on the sheet, where the deadline is also indicated. This is not mandatory and it will not count towards your mark at the exam, however, take up is strongly encouraged.
- First example sheet. (Updated: 10 November 2024.)
- First example sheet with solutions. (Updated: 20 December.)
- Second example sheet. (Updated: 20 December 2024.)
- Second example sheet with hints. (Updated: 20 December 2024.)
- Second example sheet with solutions.
- Third example sheet. (Updated: 19 November 2024.)
- Third example sheet with hints. (Updated: 19 November 2024.)
- Fourth example sheet. (Updated: 24 December 2024.)
- Fourth example sheet with hints. (Updated: 24 December 2024.)
Lecture recordings are available via Panopto. However, I expect that most students will find the notes more useful. Example classes will not be recorded.
Changelogs
Notes
- Version of 10 October 2024: Contains the introductory part, roughly the first 5 lectures.
- Version of 14 October 2024: Improved presentation of Lemma 5. Some typos corrected on pages 1-7, including a correction of the ultrametric inequality.
- Version of 18 October 2024: Some typos corrected, including in the statement of Theorem 7. Added Section 2 (Heights) and Section 3 (Proof of Dyson's Diophantine exponent).
- Version of 21 October 2024: Expanded on the comment about the optimality of Proposition 8.
- Version of 10 November 2024: Addded Section 4 (Proof of Gelfond-Scheider). Many typos corrected, mostly in Section 3.
- Version of 20 November 2024: Addded Section 5 (Integral points on curves). Some typos corrected in Section 4.
- Version of 25 November 2024: Addded Sections 6 and 7 (Multiplicative order of \(\{2,3\}\), Effective Diophantine approximation). Some typos corrected in Section 5.
- Version of 2 December 2024: Many typos corrected mostly in Section 6.
- Version of 4 December 2024: Corrected a mistake at the end of the proof of Theorem 47 (integral points on curves). The statement of Lemma 54 has a related change. Made the designation of non-examinable material clear throughout the notes. Added an explanation about what this means at the beginning.
- Version of 20 December: Corrected the bound one may obtain using Theorem 16 for effective Diophantine approximation in the last line of the notes. Fixed some typos in the proof of Theorem 56.
Example sheet 1
- Version of 24 October 2024: Correction to Q5: The point \((y_1,\ldots,y_n)\) that you need to find should be integers, not only rationals. Correction to Q11, first part: the constant \(1/d\) need to be replaced by a constant depending on \(\alpha\).
- Version of 10 November 2024: Minor typo in Q2 corrected (d replaced by n). Added solutions of Q1-8. More to follow. Has not been thoroughly checked, may contain errors.
- Version of 20 December 2024: Added solutions of remaining questions. Has not been thoroughly checked, may contain errors.
Example sheet 2
- Version of 3 November 2024: Corrected the designation of the questions to be submitted and the deadline.
- Version of 6 November 2024: Added a new question at the end of the sheet.
- Version of 15 November 2024: Some clarifications and corrections to Q3, Q4, Q6, Q11.
- Version of 20 December 2024: Clarification in Q11, that \(\varepsilon\) may be assumed small. Added draft solutions of all questions. Has not been thoroughly checked, may contain errors.
Example sheet 3
- Version of 19 November 2024: Added a new question at the end of the sheet.
Example sheet 4
- Version of 24 December 2024: Fixed typo in Q2 and clarified quantifiers.