Abstract. We show that a conceptually simple covering technique has surprisingly rich applications to density theorems and conjectures on patterns in sets involving set differences. These applications fall into three categories: (i) analogues of these statements to distance-2 versions of the pattern, (ii) reduction of these statements to relative versions, and (iii) reductions of these statements to a quasirandom case with respect to some quantities that affect the number of realisations of the pattern.

Let $d \ge 3$ be an integer. We show that whenever an order-$d$ tensor admits $d+1$ decompositions according to Tao's slice rank, if the linear subspaces spanned by their one-variable functions constitute a sunflower for each choice of special coordinate, then the tensor admits a decomposition where these linear subspaces are contained in the centers of these respective sunflowers. As an application, we deduce that for every nonnegative integer $k$ and every finite field $\mathbb{F}$ there exists an integer $C(d,k,|\mathbb{F}|)$ such that every order-$d$ tensor with slice rank $k$ over $\mathbb{F}$ admits at most $C(d,k,|\mathbb{F}|)$ decompositions with length $k$, up to a class of transformations that can be easily described.