# On the density or measure of sets and their sumsets in the integers or the circle

## TL;DR

This paper investigates the possible densities and measure properties of sets of integers and their sumsets, providing solutions to existence questions and exploring diophantine constraints within random sets and the circle.

## Contribution

It characterizes the existence of integer sequences with prescribed densities and sumset densities, and analyzes the structure of these sets in both integers and the circle, introducing diophantine constraints.

## Key findings

- Existence of integer sequences with specified density and sumset density.
- Characterization of k-tuples of densities for sets in integers and the circle.
- Introduction of diophantine constraints within pseudo s-th power sets.

## Abstract

Let $\mathrm{d}(A)$ be the asymptotic density (if it exists) of a sequence of integers $A$. For any real numbers $0\leq\alpha\leq\beta\leq 1$, we solve the question of the existence of a sequence $A$ of positive integers such that $\mathrm{d}(A)=\alpha$ and $\mathrm{d}(A+A)=\beta$. More generally we study the set of $k$-tuples $(\mathrm{d}(iA))_{1\leq i\leq k}$ for $A\subset \mathbb{N}$. This leads us to introduce subsets defined by diophantine constraints inside a random set of integers known as the set of ``pseudo $s$th powers''. We consider similar problems for subsets of the circle $\mathbb{R}/\mathbb{Z}$, that is, we partially determine the set of $k$-tuples $(\mu(iA))_{1\leq i\leq k}$ for $A\subset \mathbb{R}/\mathbb{Z}$.