The inverse conjecture for the Gowers norm over finite fields via the correspondence principle

Details The inverse conjecture for the Gowers norm over finite fields via the correspondence principle The inverse conjecture for the Gowers norm over finite fields via the correspondence principle

2008-10-30

arxiv.org/abs/0810.5527v2

Analysis & PDE Vol. 3 (2010), No. 1, 1-20

Mathematics

Combinatorics

21 pages, no figures, to appear, Analysis & PDE. This is the final version, incorporating the referee's corrections

Scientific paper

10.2140/apde.2010.3.1

The inverse conjecture for the Gowers norms $U^d(V)$ for finite-dimensional vector spaces $V$ over a finite field $\F$ asserts, roughly speaking, that a bounded function $f$ has large Gowers norm $\|f\|_{U^d(V)}$ if and only if it correlates with a phase polynomial $\phi = e_\F(P)$ of degree at most $d-1$, thus $P: V \to \F$ is a polynomial of degree at most $d-1$. In this paper, we develop a variant of the Furstenberg correspondence principle which allows us to establish this conjecture in the large characteristic case $\charac(F) \geq d$ from an ergodic theory counterpart, which was recently established by Bergelson and the authors. In low characteristic we obtain a partial result, in which the phase polynomial $\phi$ is allowed to be of some larger degree $C(d)$. The full inverse conjecture remains open in low characteristic; the counterexamples by Lovett-Meshulam-Samorodnitsky or Green-Tao in this setting can be avoided by a slight reformulation of the conjecture.

Affiliated with

Also associated with

No associations

Rating

The inverse conjecture for the Gowers norm over finite fields via the correspondence principle does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with The inverse conjecture for the Gowers norm over finite fields via the correspondence principle, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The inverse conjecture for the Gowers norm over finite fields via the correspondence principle will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-279100