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Additive Number Theory: Extremal Problems and the Combinatorics of Sumsets

Sumsets in groups

For subsets AI, ..., An of G, define the sumset

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,

Description:

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

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#Mathematics #Combinatorics #Algebra #Polynomial Method