  We prove that the slice rank of a 3-tensor (a combinatorial notion introduced
by Tao in the context of the cap-set problem), the analytic rank (a
Fourier-theoretic notion introduced by Gowers and Wolf), and the geometric rank
(a recently introduced algebro-geometric notion) are all equivalent up to an
absolute constant. As a corollary, we obtain strong trade-offs on the
arithmetic complexity of a biased bililnear map, and on the separation between
computing a bilinear map exactly and on average. Our result settles open
questions of Haramaty and Shpilka [STOC 2010], and of Lovett [Discrete Anal.,
2019] for 3-tensors.
