Small doubling in groups
Let A be a subset of a group G = (G,.). We will survey the theory of sets A with the property that |A.A| <= K|A|, where A.A = {a_1 a_2 : a_1, a_2 in A}. The case G = (Z,+) is the famous Freiman--Ruzsa theorem.
math.NT↗
arXiv subjects
Terence Tao
Publications and source records attributed to Terence Tao.
Expanding polynomials over finite fields of large characteristic, and a regularity lemma for definable sets
Let $P: \F \times \F \to \F$ be a polynomial of bounded degree over a finite field $\F$ of large characteristic. In this paper we establish the following dichotomy: either $P$ is a moderate asymmetric expander in the sense that $|P(A,B)| \gg |\F|$ whenever $A, B \subset \F$ are such that $|A| |B| \geq C |\F|^{2-1/8}$ for a sufficiently large $C$, or else $P$ takes the form $P(x,y) = Q(F(x)+G(y))$ or $P(x,y) = Q(F(x) G(y))$ for some polynomials $Q,F,G$. This is a reasonably satisfactory classification of polynomials of two variables that moderately expand (either symmetrically or asymmetrically). We obtain a similar classification for weak expansion (in which one has $|P(A,A)| \gg |A|^{1/2} |\F|^{1/2}$ whenever $|A| \geq C |\F|^{1-1/16}$), and a partially satisfactory classification for almost strong asymmetric expansion (in which $|P(A,B)| = (1-O(|\F|^{-c})) |\F|$ when $|A|, |B| \geq |\F|^{1-c}$ for some small absolute constant $c>0$). The main new tool used to establish these results is an algebraic regularity lemma that describes the structure of dense graphs generated by definable subsets over finite fields of large characteristic. This lemma strengthens the Szémeredi regularity lemma in the algebraic case, in that while the latter lemma decomposes a graph into a bounded number of components, most of which are $\eps$-regular for some small but fixed $ε$, the latter lemma ensures that all of the components are $O(|\F|^{-1/4})$-regular. This lemma, which may be of independent interest, relies on some basic facts about the étale fundamental group of an algebraic variety.
math.CO↗
The asymptotic distribution of a single eigenvalue gap of a Wigner matrix
We show that the distribution of (a suitable rescaling of) a single eigenvalue gap $λ_{i+1}(M_n)-λ_i(M_n)$ of a random Wigner matrix ensemble in the bulk is asymptotically given by the Gaudin-Mehta distribution, if the Wigner ensemble obeys a finite moment condition and matches moments with the GUE ensemble to fourth order. This is new even in the GUE case, as prior results establishing the Gaudin-Mehta law required either an averaging in the eigenvalue index parameter $i$, or fixing the energy level $u$ instead of the eigenvalue index. The extension from the GUE case to the Wigner case is a routine application of the Four Moment Theorem. The main difficulty is to establish the approximate independence of the eigenvalue counting function $N_{(-\infty,x)}(\tilde M_n)$ (where $\tilde M_n$ is a suitably rescaled version of $M_n$) with the event that there is no spectrum in an interval $[x,x+s]$, in the case of a GUE matrix. This will be done through some general considerations regarding determinantal processes given by a projection kernel.
math.PR↗
Every odd number greater than 1 is the sum of at most five primes
We prove that every odd number $N$ greater than 1 can be expressed as the sum of at most five primes, improving the result of Ramaré that every even natural number can be expressed as the sum of at most six primes. We follow the circle method of Hardy-Littlewood and Vinogradov, together with Vaughan's identity; our additional techniques, which may be of interest for other Goldbach-type problems, include the use of smoothed exponential sums and optimisation of the Vaughan identity parameters to save or reduce some logarithmic losses, the use of multiple scales following some ideas of Bourgain, and the use of Montgomery's uncertainty principle and the large sieve to improve the $L^2$ estimates on major arcs. Our argument relies on some previous numerical work, namely the verification of Richstein of the even Goldbach conjecture up to $4 \times 10^{14}$, and the verification of van de Lune and (independently) of Wedeniwski of the Riemann hypothesis up to height $3.29 \times 10^9$.
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Localisation and compactness properties of the Navier-Stokes global regularity problem
In this paper we establish a number of implications between various qualitative and quantitative versions of the global regularity problem for the Navier-Stokes equations, in the periodic, smooth finite energy, smooth $H^1$, Schwartz, or mild $H^1$ categories, and with or without a forcing term. In particular, we show that if one has global well-posedness in $H^1$ for the periodic Navier-Stokes problem with a forcing term, then one can obtain global regularity both for periodic and for Schwartz initial data (thus yielding a positive answer to both official formulations of the problem for the Clay Millennium Prize), and can also obtain global smooth solutions from smooth $H^1$ data, and global almost smooth solutions from smooth finite energy data. Our main new tools are localised energy and enstrophy estimates to the Navier-Stokes equation that are applicable for large data or long times, and which may be of independent interest.
math.AP↗
New bounds for Szemeredi's theorem, Ia: Progressions of length 4 in finite field geometries revisited
Let p > 4 be a prime. We show that the largest subset of F_p^n with no 4-term arithmetic progressions has cardinality << N(log N)^{-c}, where c = 2^{-22} and N := p^n. A result of this type was claimed in a previous paper by the authors and published in Proc. London Math. Society. Unfortunately the proof had a gap, and we issue an erratum for that paper here. Our new argument is different and significantly shorter. In fact we prove a stronger result, which can be viewed as a quantatitive version of some previous results of Bergelson-Host-Kra and the authors.
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Noncommutative sets of small doubling
A corollary of Kneser's theorem, one sees that any finite non-empty subset $A$ of an abelian group $G = (G,+)$ with $|A + A| \leq (2-\eps) |A|$ can be covered by at most $\frac{2}{\eps}-1$ translates of a finite group $H$ of cardinality at most $(2-\eps)|A|$. Using some arguments of Hamidoune, we establish an analogue in the noncommutative setting. Namely, if $A$ is a finite non-empty subset of a nonabelian group $G = (G,\cdot)$ such that $|A \cdot A| \leq (2-\eps) |A|$, then $A$ is either contained in a right-coset of a finite group $H$ of cardinality at most $\frac{2}{\eps}|A|$, or can be covered by at most $\frac{2}{\eps}-1$ right-cosets of a finite group $H$ of cardinality at most $|A|$. We also note some connections with some recent work of Sanders and of Petridis.
math.CO↗
A central limit theorem for the determinant of a Wigner matrix
We establish a central limit theorem for the log-determinant $\log|\det(M_n)|$ of a Wigner matrix $M_n$, under the assumption of four matching moments with either the GUE or GOE ensemble. More specifically, we show that this log-determinant is asymptotically distributed like $N(\log \sqrt{n!} - 1/2 \log n, 1/2 \log n)_\R$ when one matches moments with GUE, and $N(\log \sqrt{n!} - 1/4 \log n, 1/4 \log n)_\R$ when one matches moments with GOE.
math.PR↗
Random matrices: The Universality phenomenon for Wigner ensembles
In this paper, we survey some recent progress on rigorously establishing the universality of various spectral statistics of Wigner Hermitian random matrix ensembles, focusing on the Four Moment Theorem and its refinements and applications, including the universality of the sine kernel and the Central limit theorem of several spectral parameters. We also take the opportunity here to issue some errata for some of our previous papers in this area.
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Nonlinear Fourier Analysis
The nonlinear Fourier transform discussed in these notes is the map from the potential of a one dimensional discrete Dirac operator to the transmission and reflection coefficients thereof. Emphasis is on this being a nonlinear variant of the classical Fourier series, and on nonlinear analogues of classical analytic facts about Fourier series. These notes are a summary of a series of lectures given in 2003 at the Park City Mathematics Institute.
math.CA↗
The Gaussian primes contain arbitrarily shaped constellations
We show that the Gaussian primes $P[i] \subseteq \Z[i]$ contain infinitely constellations of any prescribed shape and orientation. More precisely, given any distinct Gaussian integers $v_0,...,v_{k-1}$, we show that there are infinitely many sets $\{a+rv_0,...,a+rv_{k-1}\}$, with $a \in \Z[i]$ and $r \in \Z \backslash \{0\}$, all of whose elements are Gaussian primes. The proof is modeled on a recent paper by Green and Tao and requires three ingredients. The first is a hypergraph removal lemma of Gowers and Rödl-Skokan; this hypergraph removal lemma can be thought of as a generalization of the Szemerédi-Furstenberg-Katznelson theorem concerning multidimensional arithmetic progressions. The second ingredient is the transference argument of Green and Tao, which allows one to extend this hypergraph removal lemma to a relative version, weighted by a pseudorandom measure. The third ingredient is a Goldston-Yildirim type analysis for the Gaussian integers, which yields a pseudorandom measure which is concentrated on Gaussian "almost primes".
math.CO↗
A nilpotent Freiman dimension lemma
We prove that a K-approximate subgroup of an arbitrary torsion-free nilpotent group can be covered by a bounded number of cosets of a nilpotent subgroup of bounded rank, where the bounds are explicit and depend only on K. The result can be seen as a nilpotent analogue to Freiman's dimension lemma.
math.CO↗
Random matrices: The Four Moment Theorem for Wigner ensembles
We survey some recent progress on rigorously establishing the universality of various spectral statistics of Wigner random matrix ensembles, focusing in particular on the Four Moment Theorem and its applications.
math.PR↗
Product set estimates for non-commutative groups
We develop the Plünnecke-Ruzsa and Balog-Szemerédi-Gowers theory of sum set estimates in the non-commutative setting, with discrete, continuous, and metric entropy formulations of these estimates. We also develop a Freiman-type inverse theorem for a special class of 2-step nilpotent groups, namely the Heisenberg groups with no 2-torsion in their centre.
math.CO↗
The structure of approximate groups
Let K >= 1 be a parameter. A K-approximate group is a finite set A in a (local) group which contains the identity, is symmetric, and such that A^2 is covered by K left translates of A. The main result of this paper is a qualitative description of approximate groups as being essentially finite-by-nilpotent, answering a conjecture of H. Helfgott and E. Lindenstrauss. This may be viewed as a generalisation of the Freiman-Ruzsa theorem on sets of small doubling in the integers to arbitrary groups. We begin by establishing a correspondence principle between approximate groups and locally compact (local) groups that allows us to recover many results recently established in a fundamental paper of Hrushovski. In particular we establish that approximate groups can be approximately modeled by Lie groups. To prove our main theorem we apply some additional arguments essentially due to Gleason. These arose in the solution of Hilbert's fifth problem in the 1950s. Applications of our main theorem include a finitary refinement of Gromov's theorem, as well as a generalized Margulis lemma conjectured by Gromov and a result on the virtual nilpotence of the fundamental group of Ricci almost nonnegatively curved manifolds.
math.GR↗
The Wigner-Dyson-Mehta bulk universality conjecture for Wigner matrices
A well known conjecture of Wigner, Dyson, and Mehta asserts that the (appropriately normalized) $k$-point correlation functions of the eigenvalues of random $n \times n$ Wigner matrices in the bulk of the spectrum converge (in various senses) to the $k$-point correlation function of the Dyson sine process in the asymptotic limit $n \to \infty$. There has been much recent progress on this conjecture, in particular it has been established under a wide variety of decay, regularity, and moment hypotheses on the underlying atom distribution of the Wigner ensemble, and using various notions of convergence. Building upon these previous results, we establish new instances of this conjecture with weaker hypotheses on the atom distribution and stronger notions of convergence. In particular, assuming only a finite moment condition on the atom distribution, we can obtain convergence in the vague sense, and assuming an additional regularity condition, we can upgrade this convergence to locally $L^1$ convergence.
math.PR↗
The inverse conjecture for the Gowers norm over finite fields in low characteristic
We establish the \emph{inverse conjecture for the Gowers norm over finite fields}, which asserts (roughly speaking) that if a bounded function $f: V \to \C$ on a finite-dimensional vector space $V$ over a finite field $\F$ has large Gowers uniformity norm $\|f\|_{U^{s+1}(V)}$, then there exists a (non-classical) polynomial $P: V \to \T$ of degree at most $s$ such that $f$ correlates with the phase $e(P) = e^{2πi P}$. This conjecture had already been established in the "high characteristic case", when the characteristic of $\F$ is at least as large as $s$. Our proof relies on the weak form of the inverse conjecture established earlier by the authors and Bergelson, together with new results on the structure and equidistribution of non-classical polynomials, in the spirit of the work of Green and the first author and of Kaufman and Lovett.
math.CO↗
Random matrices: Localization of the eigenvalues and the necessity of four moments
Consider the eigenvalues $λ_i(M_n)$ (in increasing order) of a random Hermitian matrix $M_n$ whose upper-triangular entries are independent with mean zero and variance one, and are exponentially decaying. By Wigner's semicircular law, one expects that $λ_i(M_n)$ concentrates around $γ_i \sqrt n$, where $\int_{-\infty}^{γ_i} ρ_{sc} (x) dx = \frac{i}{n}$ and $ρ_{sc}$ is the semicircular function. In this paper, we show that if the entries have vanishing third moment, then for all $1\le i \le n$ $$\E |λ_i(M_n)-\sqrt{n} γ_i|^2 = O(\min(n^{-c} \min(i,n+1-i)^{-2/3} n^{2/3}, n^{1/3+\eps})) ,$$ for some absolute constant $c>0$ and any absolute constant $\eps>0$. In particular, for the eigenvalues in the bulk ($\min \{i, n-i\}=Θ(n)$), $$\E |λ_i(M_n)-\sqrt{n} γ_i|^2 = O(n^{-c}). $$ \noindent A similar result is achieved for the rate of convergence. As a corollary, we show that the four moment condition in the Four Moment Theorem is necessary, in the sense that if one allows the fourth moment to change (while keeping the first three moments fixed), then the \emph{mean} of $λ_i(M_n)$ changes by an amount comparable to $n^{-1/2}$ on the average. We make a precise conjecture about how the expectation of the eigenvalues vary with the fourth moment.
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