The arithmetic removal conjecture for linear systems

Let $r,m$ be integers with $r \leqslant m$, and let $L$ be an $r \times m$ integer matrix of rank $r$. For $N$ a positive integer, write $[N]=\{1,\ldots,N\}$; a set $A\subseteq[N]$ is $L$-free if there is no vector $x\in A^m$ such that $Lx=0$.

Arithmetic removal conjecture. If $A\subseteq[N]$ and the number of vectors $x\in A^m$ satisfying $Lx=0$ is $o(N^{m-r})$, then

$$A=B\cup C,$$

where $B$ is $L$-free and $|C|=o(N)$.

This conjecture asserts that a set containing only a negligible number of solutions to the homogeneous linear system $Lx=0$ can be made $L$-free by deleting a negligible number of elements. The source presents it as a consequence that might follow from a suitably formulated higher arithmetic regularity lemma; its resolution is not given here.