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 is non-trivially correlated to 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 d = 4, by presenting an explicit function whose 4-th Gowers norm is non-negligible, but whose correlation to any polynomial of degree 3 is exponentially small. Essentially the same result (with different correlation bounds) was independently obtained by Green and Tao (2009).
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 is non-trivially correlated to 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 d = 4, by presenting an explicit function whose 4-th Gowers norm is non-negligible, but whose correlation to any polynomial of degree 3 is exponentially small. Essentially the same result (with different correlation bounds) was independently obtained by Green and Tao (2009).
| Original language | English |
|---|---|
| Article number | 1 |
| Pages (from-to) | 131-145 |
| Number of pages | 15 |
| Journal | Theory of Computing |
| Volume | 7 |
| Issue number | 1 |
| DOIs | |
| State | Published - 12 Jul 2011 |
Keywords
- Inverse Gowers conjecture
- Additive combinatorics
- Gowers norm