Published online by Cambridge University Press: 15 December 2021
In this paper we characterize when non-classical polynomials are necessary in the inverse theorem for the Gowers
$U^k$
-norm. We give a brief deduction of the fact that a bounded function on
$\mathbb F_p^n$
with large
$U^k$
-norm must correlate with a classical polynomial when
$k\le p+1$
. To the best of our knowledge, this result is new for
$k=p+1$
(when
$p>2$
). We then prove that non-classical polynomials are necessary in the inverse theorem for the Gowers
$U^k$
-norm over
$\mathbb F_p^n$
for all
$k\ge p+2$
, completely characterising when classical polynomials suffice.
Berger, Sah, Sawhney, and Tidor were supported by NSF Graduate Research Fellowship Program DGE-1745302.