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A new norm related to the Gowers U^3 norm Add to your list(s) Download to your calendar using vCal Pablo Candela Pokorna Monday 16 February 2009, 16:00-17:00 MR12, CMS, Wilberforce Road, Cambridge, CB3 0WB. If you have a question about this talk, please contact Anton Evseev. The uniformity norms (or U^d norms, for d>1 a positive integer) were introduced about ten years ago by Gowers in his effective proof of Szemerédi’s theorem, and have played an important role in arithmetic combinatorics ever since. The U^2 norm is naturally related to Fourier analysis, and a very active trend in current research aims to develop an analogue of Fourier analysis for each U^d norm with d>2. The body of results of this research for d=3 is known as quadratic Fourier analysis. After an introduction to this area we will consider a new norm related to the U^3 norm, and discuss some of its applications in quadratic Fourier analysis, including a strengthening of a central theorem of Green and Tao (the inverse theorem for the U^3 norm), and how this stronger version of the theorem can be used to give a new proof of a recent decomposition-theorem of Gowers and Wolf. This talk is part of the Junior Algebra/Combinatorics/Number Theory seminar series.

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

In the second lecture (based on Gowers’s joint work with Julia Wolf) we were introduced to decomposition theorems. A decomposition theorem for the norm can be stated as follows: if is a function (on either or ) with , there is a decomposition , where are “generalized quadratic phase functions” and are error terms with and small. This can be deduced from the inverse theorem of Green-Tao; in fact a similar statement was already implicit in their work, based on the energy increment argument. Tim presented a different approach to deducing decomposition theorems from inverse theorems, based on functional-analytic arguments involving the geometry of normed spaces (specifically, a variant of the Hahn-Banach theorem).

This can be applied to the question of counting solutions to systems of linear equations in sets. Let’s say that we are interested in finding sensible conditions under which a set will have the “statistically correct” number of solutions to a system of linear equations. For instance, if it is 4-term arithmetic progressions that we are concerned with, then uniformity is sufficient (and, in general, necessary). Green and Tao prove a more general result of this type: they define the complexity of a system of linear forms, and prove that systems of complexity are controlled by norms.

Gowers and Wolf, however, do not stop there. Suppose that, instead of 4-term progressions, we are interested in configurations of the form, say, . The complexity of this system in the sense of Green-Tao is 2, hence a set uniform in the norm will contain the “right” number of such configurations. Gowers and Wolf, however, can prove that uniformity already guarantees the same conclusion! The difference between the two examples? The squares are linearly dependent, whereas are not. Gowers and Wolf prove that such “square independence” is in fact both sufficient and necessary for a system of complexity 2 to be controlled by the $U^2$ norm. The proof is based on the decomposition theorem described earlier.

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 $d = p^2$.

We give an explicit construction of pseudorandom generators against low degree polynomials over finite fields. We show that the sum of 2d small-biased generators with error ε2O(d) is a pseudorandom generator against degree d polynomials with error ε. This gives a generator with seed length 2O(d) log(n/ε). Our construction follows the recent breakthrough result of Bogadnov and Viola. Their work shows that the sum of d small-biased generators is a pseudo-random generator against degree d polynomials, assuming the Inverse Gowers Conjecture. However, this conjecture is only proven for d=2,3. The main advantage of our work is that it does not rely on any unproven conjectures.

\F_p^N

The paper itself is called “Hypergraph regularity and the multidimensional Szemerédi theorem.” At the bottom level, the basic idea of the paper is due to Ruzsa, Szemerédi and Rödl. Ruzsa and Szemerédi started the ball rolling with a short and very clever argument that showed that Szemerédi’s famous theorem on arithmetic progressions, in the case of progressions of length 3, could be deduced from Szemerédi’s almost as famous regularity lemma, a remarkable result that allows any graph to be partitioned into a bounded number of pieces, almost all of which “behave randomly.”

trick, that can be used in many mathematical situations. With such tricks, it is usually difficult, and in any case not desirable, to formalize them as lemmas: if you try to do so then almost certainly your formal lemma will not apply in all the situations where the trick does.

Of course, in many cases, the devil really is in the details, but nevertheless knowing the overall strategy of proof is extremely valuable when trying to read that proof.

Yong-Hui Says: November 3, 2008 at 5:57 pm | Reply I am in MSRI for the cofference discrete Rigity. Green will give the first lecture. I just happen to find a question for that tricki wiki: Whether is there a common-shared refference system for that tricki wiki? Similar to that of Mathscinet of ams math review It will be a basic instrument for a mathematical website.