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Non-classical polynomials and the inverse theorem

Published online by Cambridge University Press:  15 December 2021

AARON BERGER ,
ASHWIN SAH ,
MEHTAAB SAWHNEY  and
JONATHAN TIDOR
AARON BERGER
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A e-mails: bergera@mit.edu, asah@mit.edu, msawhney@mit.edu, jtidor@mit.edu
ASHWIN SAH
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A e-mails: bergera@mit.edu, asah@mit.edu, msawhney@mit.edu, jtidor@mit.edu
MEHTAAB SAWHNEY
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A e-mails: bergera@mit.edu, asah@mit.edu, msawhney@mit.edu, jtidor@mit.edu
JONATHAN TIDOR
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A e-mails: bergera@mit.edu, asah@mit.edu, msawhney@mit.edu, jtidor@mit.edu
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Abstract

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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 173 , Issue 3 , November 2022 , pp. 525 - 537
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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Footnotes

†

Berger, Sah, Sawhney, and Tidor were supported by NSF Graduate Research Fellowship Program DGE-1745302.

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