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arXiv · 2107.07495

Non-classical polynomials and the inverse theorem

Aaron Berger·Ashwin Sah·Mehtaab Sawhney·Jonathan Tidor
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Abstract

In this note 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 $k\leq p+1$. 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 $k\geq p+2$, completely characterizing when classical polynomials suffice.

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Aaron Berger, Ashwin Sah, Mehtaab Sawhney, Jonathan Tidor. 2021-07-15. Non-classical polynomials and the inverse theorem. https://doi.org/10.1017/s0305004121000682

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