Sparse Approximation in Lattices and Semigroups

Given an integer or a non-negative integer solution $x$ to a system $Ax = b$, where the number of non-zero components of $x$ is at most $n$. This paper addresses the following question: How closely can we approximate $b$ with $Ay$, where $y$ is an integer or non-negative integer solution constrained to have at most $k$ non-zero components with $k<n$? We establish upper and lower bounds for this question in general. In specific cases, these bounds match. The key finding is that the quality of the approximation increases exponentially as $k$ goes to $n$.