# An extremal problem for integer sparse recovery

## TL;DR

This paper investigates the maximum size of integer matrices with bounded entries such that all their square submatrices are invertible, providing new bounds and solving a related geometric covering problem.

## Contribution

It introduces new bounds on the size of such matrices and addresses a specific case of a grid covering problem in higher dimensions.

## Key findings

- Established new upper bounds on matrix size d(m,k)
- Derived lower bounds for the extremal problem
- Solved a special case of the grid covering problem

## Abstract

Motivated by the problem of integer sparse recovery we study the following question. Let $A$ be an $m \times d$ integer matrix whose entries are in absolute value at most $k$. How large can be $d=d(m,k)$ if all $m \times m$ submatrices of $A$ are non-degenerate? We obtain new upper and lower bounds on $d$ and answer a special case of the problem by Brass, Moser and Pach on covering $m$-dimensional $k \times \cdots\times k$ grid by linear subspaces.