Publications (245)233.54 Total impact

Article: Chains of large gaps between primes
[Show abstract] [Hide abstract]
ABSTRACT: Let $p_n$ denote the $n$th prime, and for any $k \geq 1$ and sufficiently large $X$, define the quantity $$ G_k(X) := \max_{p_{n+k} \leq X} \min( p_{n+1}p_n, \dots, p_{n+k}p_{n+k1} ),$$ which measures the occurrence of chains of $k$ consecutive large gaps of primes. Recently, with Green and Konyagin, the authors showed that \[ G_1(X) \gg \frac{\log X \log \log X\log\log\log\log X}{\log \log \log X}\] for sufficiently large $X$. In this note, we combine the arguments in that paper with the Maier matrix method to show that \[ G_k(X) \gg \frac{1}{k^2} \frac{\log X \log \log X\log\log\log\log X}{\log \log \log X}\] for any fixed $k$ and sufficiently large $X$. The implied constant is effective and independent of $k$. 
Article: Multiple recurrence and convergence results associated to $${\text{F}}_P^\omega $$ actions
[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 (Formula Presented.). From this we deduce multiple Khintchinetype recurrence results analogous to those for ℤsystems obtained by Bergelson, Host, and Kra, and also present some new counterexamples in this setting.  [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 superpolynomial bound on the probability that a random graph has simple spectrum, along with several applications. 
Article: Long gaps between primes
[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. 
Article: Random matrices have simple spectrum
[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 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: The ElliottHalberstam conjecture implies the Vinogradov least quadratic nonresidue conjecture
[Show abstract] [Hide abstract]
ABSTRACT: For each prime $p$, let $n(p)$ denote the least quadratic nonresidue modulo $p$. Vinogradov conjectured that $n(p) = O(p^\eps)$ for every fixed $\eps>0$. This conjecture follows from the generalised Riemann hypothesis, but remains open in general. In this paper we show that Vinogradov's conjecture follows from the ElliottHalberstam conjecture on the distribution of primes in arithmetic progressions. We also give a variant of this argument that obtains bounds on short centred character sums from "Type II" estimates of the type introduced recently by Zhang and improved upon by the Polymath project. In particular, we can obtain an improvement over the Burgess bound would be obtained if one had Type II estimates with level of distribution above $2/3$ (when the conductor is not cubefree) or $3/4$ (if the conductor is cubefree). 
Article: Narrow progressions in the primes
[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 $P_i = (i1)\mathbf{m}$, 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.  [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 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.  [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 BourgainGamburd method and on the main result of our companion paper, establishing strongly dense subgroups in simple algebraic groups.  [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 ξi and $\tilde {\xi }_i$ are iid random variables that match moments to second order, the coefficients ci are deterministic, and the degree parameter n is large. Our results show, under some light conditions on the coefficients ci 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.  [Show abstract] [Hide abstract]
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.  [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 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
[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 <= KA, where A.A = {a_1 a_2 : a_1, a_2 in A}. The case G = (Z,+) is the famous FreimanRuzsa theorem.  [Show abstract] [Hide abstract]
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 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: 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 $A B \geq C \F^{21/8}$ 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 $A \geq C \F^{11/16}$), and a partially satisfactory classification for almost strong asymmetric expansion (in which $P(A,B) = (1O(\F^{c})) \F$ when $A, B \geq \F^{1c}$ 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 Sz\'emeredi 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 $\epsilon$, 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 \'etale fundamental group of an algebraic variety.  [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 complexitysstep 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 HardyLittlewood prime tuples conjecture for any linear system of finite complexity. 
Article: On Sets Defining Few Ordinary Lines
[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 "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.  [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.  [Show abstract] [Hide abstract]
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.
Publication Stats
27k  Citations  
233.54  Total Impact Points  
Top Journals
Institutions

19972014

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


2012

Yale University
New Haven, Connecticut, United States


2006

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


2003

University of Toronto
Toronto, Ontario, Canada


2001

University of Missouri
 Department of Mathematics
Columbia, Missouri, United States 
Princeton University
 Department of Mathematics
Princeton, New Jersey, United States
