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 $n$.

翻译：给定系统 $Ax = b$ 的一个整数或非负整数解 $x$，其中 $x$ 的非零分量个数至多为 $n$。本文探讨以下问题：若 $y$ 是一个整数或非负整数解，且其非零分量个数至多为 $k$（其中 $k<n$），则我们能用 $Ay$ 逼近 $b$ 到何种程度？我们针对该问题建立了普适的上界与下界。在特定情形下，这些界是匹配的。关键发现是：逼近质量随 $k$ 趋近于 $n$ 呈指数级提升。