234 research outputs found
Inverse Conjecture for the Gowers norm is false
Let be a fixed prime number, and be a large integer. The 'Inverse
Conjecture for the Gowers norm' states that if the "-th Gowers norm" of a
function f:\F_p^N \to \F_p is non-negligible, that is larger than a constant
independent of , then can be non-trivially approximated by a degree
polynomial. The conjecture is known to hold for and for any prime
. In this paper we show the conjecture to be false for and for , 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 , for .Comment: 20 page
Linear forms and higher-degree uniformity for functions on
In [GW09a] we conjectured that uniformity of degree is sufficient to
control an average over a family of linear forms if and only if the th
powers of these linear forms are linearly independent. In this paper we prove
this conjecture in , provided only that is sufficiently
large. This result represents one of the first applications of the recent
inverse theorem for the norm over by Bergelson, Tao and
Ziegler [BTZ09,TZ08]. We combine this result with some abstract arguments in
order to prove that a bounded function can be expressed as a sum of polynomial
phases and a part that is small in the appropriate uniformity norm. The precise
form of this decomposition theorem is critical to our proof, and the theorem
itself may be of independent interest.Comment: 40 page
An equivalence between inverse sumset theorems and inverse conjectures for the U^3 norm
We establish a correspondence between inverse sumset theorems (which can be
viewed as classifications of approximate (abelian) groups) and inverse theorems
for the Gowers norms (which can be viewed as classifications of approximate
polynomials). In particular, we show that the inverse sumset theorems of
Freiman type are equivalent to the known inverse results for the Gowers U^3
norms, and moreover that the conjectured polynomial strengthening of the former
is also equivalent to the polynomial strengthening of the latter. We establish
this equivalence in two model settings, namely that of the finite field vector
spaces F_2^n, and of the cyclic groups Z/NZ.
In both cases the argument involves clarifying the structure of certain types
of approximate homomorphism.Comment: 23 page
- …