[Show abstract][Hide abstract] 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 super-polynomial
bound on the probability that a random graph has simple spectrum, along with
several applications.
[Show abstract][Hide abstract] 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.
[Show abstract][Hide abstract] ABSTRACT: Let $M_n = (\xi_{ij})_{1 \leq i,j \leq n}$ be a real symmetric random matrix
in which the upper-triangular 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}s-Renyi random graph has simple spectrum asymptotically almost surely,
answering a question of Babai.
[Show abstract][Hide abstract] 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 $P_i = (i-1)\mathbf{m}$, 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.
[Show abstract][Hide abstract] 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.
[Show abstract][Hide abstract] 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
1-parameter 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 1-parameter 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 · 3.24 Impact Factor
[Show abstract][Hide abstract] 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
Bourgain-Gamburd method and on the main result of our companion paper,
establishing strongly dense subgroups in simple algebraic groups.
Journal of the European Mathematical Society 09/2013; 17(6). DOI:10.4171/JEMS/533 · 1.70 Impact Factor
[Show abstract][Hide abstract] 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
non-concentration bounds are obeyed.
International Mathematics Research Notices 07/2013; 2015(13). DOI:10.1093/imrn/rnu084 · 1.10 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We establish a version of the Furstenberg-Katznelson multi-dimensional
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; 207(1). DOI:10.1007/s11856-015-1157-9 · 0.79 Impact Factor
[Show abstract][Hide abstract] 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 Khintchine-type recurrence
results analogous to those for $\Z$-systems obtained by Bergelson, Host, and
Kra, and also present some new counterexamples in this setting.
[Show abstract][Hide abstract] 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| <= 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.
[Show abstract][Hide abstract] ABSTRACT: We establish a new mixing theorem for quasirandom groups (finite groups with
no low-dimensional 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
non-elementary, 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.
[Show abstract][Hide abstract] ABSTRACT: We prove the inverse conjecture for the Gowers U s+1[N]-norm for all s ≥ 1; this is new for s ≥ 4. More precisely, we establish that if f: [N] → [-1; 1] is a function with ||f|| Us+1[N] ≥ δ, then there is a bounded complexitys-step nilsequence F(g(n)Γ) that correlates with f, where the bounds on the complexity and correlation depend only on s and δ. From previous results, this conjecture implies the Hardy-Littlewood prime tuples conjecture for any linear system of finite complexity.
Annals of Mathematics 09/2012; 176(2):1231-1372. DOI:10.4007/annals.2012.176.2.11 · 3.24 Impact Factor
[Show abstract][Hide abstract] 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 "orchard-planting
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.
[Show abstract][Hide abstract] 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.
The Annals of Probability 06/2012; 43(2). DOI:10.1214/13-AOP876 · 1.42 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: 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.
[Show abstract][Hide abstract] 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.
[Show abstract][Hide abstract] 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.
[Show abstract][Hide abstract] 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 third