Publications (197)172.56 Total impact

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ABSTRACT: Gaps (or spacings) between consecutive eigenvalues are a central topic in random matrix theory. The goal of this paper is to study the tail distribution of these gaps in various random matrix models. We give the first repulsion bound for random matrices with discrete entries and the first superpolynomial bound on the probability that a random graph has simple spectrum, along with several applications. 
Article: Long gaps between primes
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ABSTRACT: Let $p_n$ denotes the $n$th prime. We prove that $$\max_{p_{n+1} \leq X} (p_{n+1}p_n) \gg \frac{\log X \log \log X\log\log\log\log X}{\log \log \log X}$$ for sufficiently large $X$, improving upon recent bounds of the first three and fifth authors and of the fourth author. Our main new ingredient is a generalization of a hypergraph covering theorem of Pippenger and Spencer, proven using the R\"odl nibble method. 
Article: Random matrices have simple spectrum
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ABSTRACT: Let $M_n = (\xi_{ij})_{1 \leq i,j \leq n}$ be a real symmetric random matrix in which the uppertriangular entries $\xi_{ij}, i<j$ and diagonal entries $\xi_{ii}$ are independent. We show that with probability tending to 1, $M_n$ has no repeated eigenvalues. As a corollary, we deduce that the Erd{\H o}sRenyi random graph has simple spectrum asymptotically almost surely, answering a question of Babai. 
Article: Narrow progressions in the primes
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ABSTRACT: In a previous paper of the authors, we showed that for any polynomials $P_1,\dots,P_k \in \Z[\mathbf{m}]$ with $P_1(0)=\dots=P_k(0)$ and any subset $A$ of the primes in $[N] = \{1,\dots,N\}$ of relative density at least $\delta>0$, one can find a "polynomial progression" $a+P_1(r),\dots,a+P_k(r)$ in $A$ with $0 < r \leq N^{o(1)}$, if $N$ is sufficiently large depending on $k,P_1,\dots,P_k$ and $\delta$. In this paper we shorten the size of this progression to $0 < r \leq \log^L N$, where $L$ depends on $k,P_1,\dots,P_k$ and $\delta$. In the linear case $P_i = (i1)\mathbf{m}$, we can take $L$ independent of $\delta$. The main new ingredient is the use of the densification method of Conlon, Fox, and Zhao to avoid having to directly correlate the enveloping sieve with dual functions of unbounded functions. 
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ABSTRACT: Let $G(X)$ denote the size of the largest gap between consecutive primes below $X$. Answering a question of Erdos, we show that $$G(X) \geq f(X) \frac{\log X \log \log X \log \log \log \log X}{(\log \log \log X)^2},$$ where $f(X)$ is a function tending to infinity with $X$. Our proof combines existing arguments with a random construction covering a set of primes by arithmetic progressions. As such, we rely on recent work on the existence and distribution of long arithmetic progressions consisting entirely of primes. 
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ABSTRACT: This is an erratum to 'On the quantitative distribution of polynomial nilsequences' [GT]. The proof of Theorem 8.6 of that paper, which claims a distribution result for multiparameter polynomial sequences on nilmanifolds, was incorrect. We provide two fixes for this issue here. First, we deduce the "equal sides" case $N_1 = \dots = N_t = N$ of [GT, Theorem 8.6] from the 1parameter results in [GT]. This is the same basic mode of argument we attempted in the original paper, though the details are different. The equal sides case is the only one required in applications such as the proof of the inverse conjectures for the Gowers norms due to the authors and Ziegler. Second, we sketch a proof that [GT, Theorem 8.6] does in fact hold in its originally stated form, that is to say without the equal sides condition. To obtain this statement the entire argument of [GT] must be run in the context of multiparameter polynomial sequences $g : \mathbb{Z}^t \rightarrow G$ rather than 1parameter sequences $g : \mathbb{Z} \rightarrow G$ as is currently done.Annals of Mathematics 11/2013; 179(3). DOI:10.4007/annals.2014.179.3.8 · 2.82 Impact Factor 
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ABSTRACT: We show that random Cayley graphs of finite simple (or semisimple) groups of Lie type of fixed rank are expanders. The proofs are based on the BourgainGamburd method and on the main result of our companion paper, establishing strongly dense subgroups in simple algebraic groups. 
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ABSTRACT: In this paper, we establish some local universality results concerning the correlation functions of the zeroes of random polynomials with independent coefficients. More precisely, consider two random polynomials $f =\sum_{i=1}^n c_i \xi_i z^i$ and $\tilde f =\sum_{i=1}^n c_i \tilde \xi_i z^i$, where the $\xi_i$ and $\tilde \xi_i$ are iid random variables that match moments to second order, the coefficients $c_i$ are deterministic, and the degree parameter $n$ is large. Our results show, under some light conditions on the coefficients $c_i$ and the tails of $\xi_i, \tilde \xi_i$, that the correlation functions of the zeroes of $f$ and $\tilde f$ are approximately the same. As an application, we give some answers to the classical question `"How many zeroes of a random polynomials are real?" for several classes of random polynomial models. Our analysis relies on a general replacement principle, motivated by some recent work in random matrix theory. This principle enables one to compare the correlation functions of two random functions $f$ and $\tilde f$ if their log magnitudes $\log f, \log\tilde f$ are close in distribution, and if some nonconcentration bounds are obeyed.International Mathematics Research Notices 07/2013; DOI:10.1093/imrn/rnu084 · 1.07 Impact Factor 
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ABSTRACT: We establish a version of the FurstenbergKatznelson multidimensional Szemer\'edi in the primes ${\mathcal P} := \{2,3,5,\ldots\}$, which roughly speaking asserts that any dense subset of ${\mathcal P}^d$ contains constellations of any given shape. Our arguments are based on a weighted version of the Furstenberg correspondence principle, relative to a weight which obeys an infinite number of pseudorandomness (or "linear forms") conditions, combined with the main results of a series of papers by Green and the authors which establish such an infinite number of pseudorandomness conditions for a weight associated with the primes. The same result, by a rather different method, has been simultaneously established by Cook, Magyar, and Titichetrakun.Israel Journal of Mathematics 06/2013; DOI:10.1007/s1185601511579 · 0.66 Impact Factor 
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ABSTRACT: Using an ergodic inverse theorem obtained in our previous paper, we obtain limit formulae for multiple ergodic averages associated with the action of $\F_{p}^{\omega}$. From this we deduce multiple Khintchinetype recurrence results analogous to those for $\Z$systems obtained by Bergelson, Host, and Kra, and also present some new counterexamples in this setting. 
Article: Small doubling in groups
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ABSTRACT: Let A be a subset of a group G = (G,.). We will survey the theory of sets A with the property that A.A <= KA, where A.A = {a_1 a_2 : a_1, a_2 in A}. The case G = (Z,+) is the famous FreimanRuzsa theorem. 
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ABSTRACT: We establish a new mixing theorem for quasirandom groups (finite groups with no lowdimensional unitary representations) $G$ which, informally speaking, asserts that if $g, x$ are drawn uniformly at random from $G$, then the quadruple $(g,x,gx,xg)$ behaves like a random tuple in $G^4$, subject to the obvious constraint that $gx$ and $xg$ are conjugate to each other. The proof is nonelementary, proceeding by first using an ultraproduct construction to replace the finitary claim on quasirandom groups with an infinitary analogue concerning a limiting group object that we call an \emph{ultra quasirandom group}, and then using the machinery of idempotent ultrafilters to establish the required mixing property for such groups. Some simpler recurrence theorems (involving tuples such as $(x,gx,xg)$) are also presented, as well as some further discussion of specific examples of ultra quasirandom groups.Geometric and Functional Analysis 11/2012; 24(1). DOI:10.1007/s0003901402520 · 1.32 Impact Factor 
Article: On sets defining few ordinary lines
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ABSTRACT: Let P be a set of n points in the plane, not all on a line. We show that if n is large then there are at least n/2 ordinary lines, that is to say lines passing through exactly two points of P. This confirms, for large n, a conjecture of Dirac and Motzkin. In fact we describe the exact extremisers for this problem, as well as all sets having fewer than n  C ordinary lines for some absolute constant C. We also solve, for large n, the "orchardplanting problem", which asks for the maximum number of lines through exactly 3 points of P. Underlying these results is a structure theorem which states that if P has at most Kn ordinary lines then all but O(K) points of P lie on a cubic curve, if n is sufficiently large depending on K.Discrete and Computational Geometry 08/2012; DOI:10.1007/s0045401395189 · 0.61 Impact Factor 
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ABSTRACT: It is a classical result of Ginibre that the normalized bulk $k$point correlation functions of a complex $n \times n$ gaussian matrix with independent entries of mean zero and unit variance are asymptotically given by the determinantal point process on $\C$ with kernel $K_\infty(z,w) := \frac{1}{\pi} e^{z^2/2  w^2/2 + z \bar{w}}$ in the limit $n \to \infty$. In this paper we show that this asymptotic law is universal among all random $n \times n$ matrices $M_n$ whose entries are jointly independent, exponentially decaying, have independent real and imaginary parts, and whose moments match that of the complex gaussian ensemble to fourth order. Analogous results at the edge of the spectrum are also obtained. As an application, we extend a central limit theorem for the number of eigenvalues of complex gaussian matrices in a small disk, to these more general ensembles. Our method also extends to the case of matrices which match the real gaussian ensemble instead of the complex one. As an application, we show that a real $n \times n$ matrix whose entries are jointly independent, exponentially decaying, and whose moments match the real gaussian ensemble to fourth order has $\sqrt{\frac{2n}{\pi}} + o(\sqrt{n})$ real eigenvalues asymptotically almost surely. 
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ABSTRACT: Let p > 4 be a prime. We show that the largest subset of F_p^n with no 4term 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 BergelsonHostKra and the authors. 
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ABSTRACT: 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. 
Article: Nonlinear Fourier Analysis
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ABSTRACT: 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. 
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ABSTRACT: Let $W_n= \frac{1}{\sqrt n} M_n$ be a Wigner matrix whose entries have vanishing third moment, normalized so that the spectrum is concentrated in the interval $[2,2]$. We prove a concentration bound for $N_I = N_I(W_n)$, the number of eigenvalues of $W_n$ in an interval $I$. Our result shows that $N_I$ decays exponentially with standard deviation at most $O(\log^{O(1)} n)$. This is best possible up to the constant exponent in the logarithmic term. As a corollary, the bulk eigenvalues are localized to an interval of width $O(\log^{O(1)} n/n)$; again, this is optimal up to the exponent. These results strengthen recent results of Erdos, Yau and Yin (under the extra assumption of vanishing third01/2012; DOI:10.1142/S201032631350007X 
01/2012; 9(4):29853059. DOI:10.4171/OWR/2012/50

Annals of Mathematics 01/2012; 176(2):12311372. DOI:10.4007/annals.2012.176.2.11 · 2.82 Impact Factor
Publication Stats
21k  Citations  
172.56  Total Impact Points  
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Institutions

1997–2012

University of California, Los Angeles
 Department of Mathematics
Los Ángeles, California, United States


2006–2007

California Institute of Technology
 Department of Electrical Engineering
Pasadena, California, United States


2005

University of Bristol
 School of Mathematics
Bristol, England, United Kingdom


2001

University of Toronto
 Department of Mathematics
Toronto, Ontario, Canada


1999

Washington University in St. Louis
 Department of Mathematics
San Luis, Missouri, United States
