Explore the foundations of additive number theory in this comprehensive lecture on extremal problems and the combinatorics of sumsets. Delve into key concepts such as sumsets in groups, density of sets, and lower bounds for sums of finite sets. Learn about important theorems like Cauchy-Davenport and Dias da Silva-Hamidoune, and their proofs using the polynomial method. Investigate extremal properties of additive bases, including the Erdos-Turan conjecture, thin bases, and minimal asymptotic bases. Gain insights into the interconnections between combinatorics, number theory, and group theory through this in-depth exploration of additive combinatorics.
Additive Number Theory - Extremal Problems and the Combinatorics of Sumsets by M. Nathanson
International Centre for Theoretical Sciences via YouTube
Overview
Syllabus
Additive Number Theory: Extremal Problems and the Combinatorics of Sumsets
Sumsets in groups
For subsets AI, ..., An of G, define the sumset
Simple questions
Simple questions
Density of sets and sumsets of integers
Lower bounds for sums of finite sets
We ave similar bounds for sets of congruence classes
There are many proofs. Here is an elementary proof that uses the "polynomial method."
Lemma
Lemma - If A and are subsets of a finite set G, then
Lemma - Let A and be subsets of a finite abelian group G.
Theorem Cauchy-Davenport
Because
Lemma
Theorem
Consider the monomial xmyn.
Theorem Dias da Silva-Hamidoune
References
Extremal properties of additive bases
Erdos-Turan conjecture
Thin bases - An asymptotic basis A of order is thin if
Minimal asymptotic bases
Idea 1970: If the Erdos-Turan conjecture were false,
Taught by
International Centre for Theoretical Sciences
