Lattice Covering Densities and Additive Combinatorics

The well-known Lattice Covering Problem asks for the most optimal way to cover the space $\mathbb{R}^n$, $n \geq 2$, by using copies of an Euclidean ball centered at points of a given lattice. More precisely, consider a closed Euclidean ball $B$ and a lattice $L \subset \mathbb{R}^d$, we say that $L$ is a covering lattice for $B$ if

\begin{equation} \label{eqn:ecuation1}

    \mathbb{R}^n = L + B

\end{equation}

The {\emph{covering density} $\displaystyle \Theta (L)$ of whole space $\mathbb{R}^n$ is defined as the minimal volume of a closed Euclidean Ball $B$ for which (\ref{eqn:ecuation1}) holds. Define

\begin{center}

    $\displaystyle \Theta_n := \inf  \left \{   \Theta (L): L \text{ is a lattice in $\mathbb{R}^n$ of covolume one} \right \} $

\end{center}

to be the minimal density of lattice coverings of $\mathbb{R}^n$. Where the covolume of $L$ is the volume of its fundamental parallepipeds (sometimes refer as the determinant of $L$). Thus the Lattice Covering Problem asks for the best upper bound for $\displaystyle \Theta_n$.

So far, this problem has only been studied geometrically using Kakeya-type methods to obtain results for convex bodies in place of balls. In this talk, we make a connection between lattice covering densities and additive combinatorics, and consider the more general setting of approximate groups and sets with low doubling or high additive energy. This is joint work with Francisco Romero Acosta.