Inverse Conjecture for the Gowers norm is false
Authors:
Shachar Lovett,
Roy Meshulam,
Alex Samorodnitsky
(Submitted on 21 Nov 2007)
Abstract: Let $p$ be a fixed prime number, and $N$ be a large integer. The 'Inverse
Conjecture for the Gowers norm' states that if the "$d$-th Gowers norm" of a
function $f:\F_p^N \to \F_p$ is non-negligible, that is larger than a constant
independent of $N$, then $f$ can be non-trivially approximated by a degree
$d-1$ polynomial. The conjecture is known to hold for $d=2,3$ and for any prime
$p$. In this paper we show the conjecture to be false for $p=2$ and for $d =
4$, by presenting an explicit function whose 4-th Gowers norm is
non-negligible, but whose correlation any polynomial of degree 3 is
exponentially small.
Essentially the same result (with different correlation bounds) was
independently obtained by Green and Tao \cite{gt07}. Their analysis uses a
modification of a Ramsey-type argument of Alon and Beigel \cite{ab} to show
inapproximability of certain functions by low-degree polynomials. We observe
that a combination of our results with the argument of Alon and Beigel implies
the inverse conjecture to be false for any prime $p$, for $d = p^2$.
Comments:
20 pages
Group items matching
in title, tags, annotations or url
3More
Conference update, part II « The Accidental Mathematician - 0 views
0809.3674v1.pdf (application/pdf 对象) - 0 views
3More
What might an expository mathematical wiki be like? « Gowers's Weblog - 0 views
Timothy Gowers's web page - 0 views
Timothy Gowers links page - 0 views
1 - 19 of 19
Showing 20▼ items per page