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

By Ben Joseph Green, Terence Tao, T. Ziegler. 2010

282 citations recorded by the source corpus.

In this note we announce the proof of the inverse conjecture for the Gowers U^{s+1}[N]-norm for all s => 3; this is new for s => 4, the cases s = 1,2,3 having been previously established. More precisely we outline a proof (details of which will appear in a forthcoming paper) that if f : [N] -> [-1,1] is a function with || f ||_{U^{s+1}[N]} => \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 3.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/1967b91ce0dcd2a910f0b74de65afa7753d52f0d

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Citations in the corpus

Long gaps between primes (2014, methodology): “The arguments in [12] used recent results [18, 19, 20] on the number of solutions to linear equations in primes, whereas the arguments in [29] instead relied on multidimensional prime-detecting sieves introduced in [27].”
Higher order Fourier analysis of multiplicative functions and applications (2014, methodology): “f Theorem 3.3 and completes the proof. Our next goal is for every s≥ 2 to replace the estimate in (3.12) with the estimate kfN,unkUs(Z Ne) ≤ ε. To this end, we shall see (using the inverse theorem in [32]) that it suﬃces to get a strengthening of (3.12) where the place of (e(n2α)) takes any s-step n”
THE LOGARITHMICALLY AVERAGED CHOWLA AND ELLIOTT CONJECTURES FOR TWO-POINT CORRELATIONS (2015, methodology): “Using the literature from higher order Fourier analysis (in particular the inverse theorem in [15], together with transference arguments from [9, 14], or [33]), one is now faced with the task of controlling sums even more complicated than (51), in which the linear phases n 7→ e(αn) are now replaced ”
The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures (2017, background): “It is only known that the Liouville function is Gowers uniform over long intervals (by [20], [21]), as opposed to short ones, and even the case of local U2-uniformity of the Liouville function is a difficult open problem (also mentioned in [40, Section 4]).”
Primes in intervals of bounded length (2014, background): “Along with Ziegler in [30], they went on to prove the prime k-tuplets conjecture for any admissible set of linear forms, provided that no two satisfy a linear equation over the integers.”
The Inverse Conjecture for the Gowers Norm over Finite Fields in Low Characteristic (2011, methodology): “rmalise this rigorously, one needs to develop a theory of polynomial maps in several variables, that generalises the material in Section C to groups ﬁltered by Nd rather than N. This can be done (see [24]), but we will not introduce this additional notation here. In the next section we will use the ”
Expander graphs in pure and applied mathematics (2011, background): “ive. Then r 0(O;f) = n, i.e., the set of x2Owhose all coordinates are simultaneously primes is Zariski dense in b+ C and in particular, it is innite. A recent breakthrough of Green, Tao and Ziegler ([GTZ], [GT2]) has proved this conjecture for the case when rank() 2. The most dicult case is when ran”