Fundamental theorems in additive number theory

Abstract: Let $A$ be a subset of the integers $\mathbf Z$, of the lattice ${\mathbf Z}^n$, or of any additive abelian semigroup $X$. The central problem in additive number theory is to understand the $h$-fold sumset \[ hA = \{a_1+\cdots + a_h : a_i \in A \text{ for all } i=1,\ldots, h \}. \] If $A$ is finite, what is the size of the sumset $hA$? If $A$ is infinite, what is the density of $hA$? What is the structure of the sumset $hA$? Describe this for small $h$, and also asymptotically as $h \rightarrow \infty$. In how many ways can an element $x \in X$ be represented as the sum of $h$ elements of $A$? For fixed $r$, what is the subset of $hA$ consisting of elements that have at least $r$ representations? Classical problems consider sums of squares, of $k$th powers, and of primes, but the general case is also important. This talk will discuss both old and very recent results about sumsets.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory (CANT 2021)

Series comments: This is the nineteenth in a series of annual workshops sponsored by the New York Number Theory Seminar on problems in combinatorial and additive number theory and related parts of mathematics.

Registration for the conference is free. Register at cant2021.eventbrite.com.

The conference website is www.theoryofnumbers.com/cant/ Lectures will be broadcast on Zoom. The Zoom login will be emailed daily to everyone who has registered on eventbrite. To join the meeting, you may need to download the free software from www.zoom.us.

The conference program, list of speakers, and abstracts are posted on the external website.