An inverse theorem for the Gowers $U^{s+1}[N]$-norm

Pages: 69 - 90,
Volume 18,
2011 doi:10.3934/era.2011.18.69

Ben Green - Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, United Kingdom (email)

Terence Tao - Department of Mathematics, UCLA, 405 Hilgard Ave, Los Angeles, CA 90095, United States (email)

Tamar Ziegler - Department of Mathematics, Technion - Israel Institute of Technology, Haifa, 32000, Israel (email)

Abstract:
This is an announcement of the proof of the *inverse conjecture for the Gowers $U^{s+1}[N]$-norm* for all $s \geq 3$; this is new for $s \geq 4$, the cases $s = 1,2,3$ having been previously established. More precisely we outline a proof that if $f : [N] \rightarrow [-1,1]$ is a function with ||$f$|| $U^{s+1}[N] \geq \delta$ then there is a bounded-complexity $s$-step nilsequence $F(g(n)\Gamma)$ which correlates with $f$, where the bounds on the complexity and correlation depend only on $s$ and $\delta$. From previous results, this conjecture implies the Hardy-Littlewood prime tuples conjecture for any linear system of finite complexity. In particular, one obtains an asymptotic formula for the number of $k$-term arithmetic progressions $p_1 < p_2 < ... < p_k \leq N$ of primes, for every $k \geq 3$.

Keywords: Gowers norms, nilsequences.

Mathematics Subject Classification: Primary: 11B99.

Received: March 2011;
Revised:
May 2011;
Available Online: July 2011.

References