The Partition Rank of a Tensor and $k$-Right Corners in ${\mathbb{F}}_q^n$

Abstract

Following the breakthrough of Croot, Lev, and Pach, Tao introduced a symmetrized version of their argument, which is now known as the slice rank method. In this paper, we introduce a more general version of the slice rank of a tensor, which we call the Partition Rank. This allows us to extend the slice rank method to problems that require the variables to be distinct. Using the partition rank, we generalize a recent result of Ge and Shangguan, and prove that any set $A\subset {\mathbb{F}}_q^n$ of size \[|A|>\binom{n+(k-1)q}{(k-1)(q-1)}\] contains a $k$-right-corner, that is distinct vectors $x_1,\dots ,x_k,x_{k+1}$ where $x_1-x_{k+1},\dots ,x_k-x_{k+1}$ are mutually orthogonal, for $q=p^r$, a prime power with $p>k$.

Updated section 5, as well as the proof for the improved k=2 bound