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Introduction to additive combinatorics lecture 1.0 --- What is additive combinatorics?
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This is an introductory video to a 16-hour course on additive combinatorics given as part of Cambridge's Part III mathematics course in the academic year 2021-2. After a few remarks about practicalities, I informally discuss a few open problems, and attempt to explain what additive combinatorics is and how it differs in emphasis from combinatorial number theory.
Notes for the course, which will be updated as I give it, are available here.
0:00 Introduction
3:35 Description of combinatorial number theory.
7:44 The Erdös additive basis problem.
14:03 The growth rate of infinite Sidon sets.
22:00 Statement of Szemerédi's theorem.
25:14 The Erdös-Szemerédi sum-product problem.
30:33 The difference between additive combinatorics and combinatorial number theory.
At around 29:33 I forget the details about the state of the art on the sum-product problem. The current best known exponent is 4/3+2/1167 and it is due to Misha Rudnev and Sophie Stevens.
Notes for the course, which will be updated as I give it, are available here.
0:00 Introduction
3:35 Description of combinatorial number theory.
7:44 The Erdös additive basis problem.
14:03 The growth rate of infinite Sidon sets.
22:00 Statement of Szemerédi's theorem.
25:14 The Erdös-Szemerédi sum-product problem.
30:33 The difference between additive combinatorics and combinatorial number theory.
At around 29:33 I forget the details about the state of the art on the sum-product problem. The current best known exponent is 4/3+2/1167 and it is due to Misha Rudnev and Sophie Stevens.
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