On the density of sumsets

Abstract: We define a large family $\mathcal{D}$ of partial set functions $\mu: \mathrm{dom}(\mu) \subseteq \mathcal{P}(\mathbf{N}) \to \mathbf{R}$ satisfying certain axioms. Examples of "densities" $\mu \in \mathcal{D}$ include the asymptotic, Banach, logarithmic, analytic, Pólya, and Buck densities. We prove several results on sumsets which were previously obtained for the asymptotic density. For instance, we show that for each $n \in \mathbf N^+$ and $\alpha \in [0,1]$, there exists $A \subseteq \mathbf{N}$ with $kA \in \text{dom}(\mu)$ and $\mu(kA) = \alpha k/n$ for every $\mu \in \mathcal{D}$ and every $k=1,\ldots, n$, where $kA$ denotes the $k$-fold sumset $A+\cdots+A$. Joint work with Salvatore Tringali.

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.