arXiv subjects

Terence Tao

Publications and source records attributed to Terence Tao.

At least 109 records · Page 6Linked to original sources

arXiv2019

The structure of correlations of multiplicative functions at almost all scales, with applications to the Chowla and Elliott conjectures

We study the asymptotic behaviour of higher order correlations $$ \mathbb{E}_{n \leq X/d} g_1(n+ah_1) \cdots g_k(n+ah_k)$$ as a function of the parameters $a$ and $d$, where $g_1,\dots,g_k$ are bounded multiplicative functions, $h_1,\dots,h_k$ are integer shifts, and $X$ is large. Our main structural result asserts, roughly speaking, that such correlations asymptotically vanish for almost all $X$ if $g_1 \cdots g_k$ does not (weakly) pretend to be a twisted Dirichlet character $n \mapsto χ(n)n^{it}$, and behave asymptotically like a multiple of $d^{-it} χ(a)$ otherwise. This extends our earlier work on the structure of logarithmically averaged correlations, in which the $d$ parameter is averaged out and one can set $t=0$. Among other things, the result enables us to establish special cases of the Chowla and Elliott conjectures for (unweighted) averages at almost all scales; for instance, we establish the $k$-point Chowla conjecture $ \mathbb{E}_{n \leq X} λ(n+h_1) \cdots λ(n+h_k)=o(1)$ for $k$ odd or equal to $2$ for all scales $X$ outside of a set of zero logarithmic density.

math.NT↗

arXiv2019

On the universality of the incompressible Euler equation on compact manifolds, II. Non-rigidity of Euler flows

The incompressible Euler equations on a compact Riemannian manifold $(M,g)$ take the form \begin{align*} \partial_t u + \nabla_u u &= - \mathrm{grad}_g p \\ \mathrm{div}_g u &= 0, \end{align*} where $u: [0,T] \to Γ(T M)$ is the velocity field and $p: [0,T] \to C^\infty(M)$ is the pressure field. In this paper we show that if one is permitted to extend the base manifold $M$ by taking an arbitrary warped product with a torus, then the space of solutions to this equation becomes "non-rigid'"in the sense that a non-empty open set of smooth incompressible flows $u: [0,T] \to Γ(T M)$ can be approximated in the smooth topology by (the horizontal component of) a solution to these equations. We view this as further evidence towards the "universal" nature of Euler flows.

math.AP↗

arXiv2019

Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges

We show that the expected asymptotic for the sums $\sum_{X < n \leq 2X} Λ(n) Λ(n+h)$, $\sum_{X < n \leq 2X} d_k(n) d_l(n+h)$, and $\sum_{X < n \leq 2X} Λ(n) d_k(n+h)$ hold for almost all $h \in [-H,H]$, provided that $X^{8/33+\varepsilon} \leq H \leq X^{1-\varepsilon}$, with an error term saving on average an arbitrary power of the logarithm over the trivial bound. Previous work of Mikawa, Perelli-Pintz and Baier-Browning-Marasingha-Zhao covered the range $H \geq X^{1/3+\varepsilon}$. We also obtain an analogous result for $\sum_n Λ(n) Λ(N-n)$. Our proof uses the circle method and some oscillatory integral estimates (following a paper of Zhan) to reduce matters to establishing some mean-value estimates for certain Dirichlet polynomials associated to "Type $d_3$" and "Type $d_4$" sums (as well as some other sums that are easier to treat). After applying Hölder's inequality to the Type $d_3$ sum, one is left with two expressions, one of which we can control using a short interval mean value theorem of Jutila, and the other we can control using exponential sum estimates of Robert and Sargos. The Type $d_4$ sum is treated similarly using the classical $L^2$ mean value theorem and the classical van der Corput exponential sum estimates.

math.NT↗

arXiv2018

Fourier uniformity of bounded multiplicative functions in short intervals on average

Let $λ$ denote the Liouville function. We show that as $X \rightarrow \infty$, $$ \int_{X}^{2X} \sup_α \left | \sum_{x < n \leq x + H} λ(n) e(-αn) \right | dx = o ( X H) $$ for all $H \geq X^θ$ with $θ> 0$ fixed but arbitrarily small. Previously, this was only known for $θ> 5/8$. For smaller values of $θ$ this is the first `non-trivial' case of local Fourier uniformity on average at this scale. We also obtain the analogous statement for (non-pretentious) $1$-bounded multiplicative functions. We illustrate the strength of the result by obtaining cancellations in the sum of $λ(n) Λ(n + h) Λ(n + 2h)$ over the ranges $h < X^θ$ and $n < X$, and where $Λ$ is the von Mangoldt function.

math.NT↗

arXiv2018

The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures

Let $g_0,\dots,g_k: {\bf N} \to {\bf D}$ be $1$-bounded multiplicative functions, and let $h_0,\dots,h_k \in {\bf Z}$ be shifts. We consider correlation sequences $f: {\bf N} \to {\bf Z}$ of the form $$ f(a):= \widetilde{\lim}_{m \to \infty} \frac{1}{\log ω_m} \sum_{x_m/ω_m \leq n \leq x_m} \frac{g_0(n+ah_0) \dots g_k(n+ah_k)}{n} $$ where $1 \leq ω_m \leq x_m$ are numbers going to infinity as $m \to \infty$, and $\widetilde{\lim}$ is a generalised limit functional extending the usual limit functional. We show a structural theorem for these sequences, namely that these sequences $f$ are the uniform limit of periodic sequences $f_i$. Furthermore, if the multiplicative function $g_0 \dots g_k$ "weakly pretends" to be a Dirichlet character $χ$, the periodic functions $f_i$ can be chosen to be $χ$-isotypic in the sense that $f_i(ab) = f_i(a) χ(b)$ whenever $b$ is coprime to the periods of $f_i$ and $χ$, while if $g_0 \dots g_k$ does not weakly pretend to be any Dirichlet character, then $f$ must vanish identically. As a consequence, we obtain several new cases of the logarithmically averaged Elliott conjecture, including the logarithmically averaged Chowla conjecture for odd order correlations. We give a number of applications of these special cases, including the conjectured logarithmic density of all sign patterns of the Liouville function of length up to three, and of the Möbius function of length up to four.

math.NT↗

arXiv2018

Commutators close to the identity

Let $D,X \in B(H)$ be bounded operators on an infinite dimensional Hilbert space $H$. If the commutator $[D,X] = DX-XD$ lies within $\varepsilon$ in operator norm of the identity operator $1_{B(H)}$, then it was observed by Popa that one has the lower bound $\| D \| \|X\| \geq \frac{1}{2} \log \frac{1}{\varepsilon}$ on the product of the operator norms of $D,X$; this is a quantitative version of the Wintner-Wielandt theorem that $1_{B(H)}$ cannot be expressed as the commutator of bounded operators. On the other hand, it follows easily from the work of Brown and Pearcy that one can construct examples in which $\|D\| \|X\| = O(\varepsilon^{-2})$. In this note, we improve the Brown-Pearcy construction to obtain examples of $D,X$ with $\| [D,X] - 1_{B(H)} \| \leq \varepsilon$ and $\| D\| \|X\| = O( \log^{5} \frac{1}{\varepsilon} )$.

math.OA↗

arXiv2018

An inverse theorem for an inequality of Kneser

Let $G = (G,+)$ be a compact connected abelian group, and let $μ_G$ denote its probability Haar measure. A theorem of Kneser (generalising previous results of Macbeath and Raikov) establishes the bound $$ μ_G(A + B) \geq \min( μ_G(A)+μ_G(B), 1 ) $$ whenever $A,B$ are compact subsets of $G$, and $A+B := \{ a+b: a \in A, b \in B \}$ denotes the sumset of $A$ and $B$. Clearly one has equality when $μ_G(A)+μ_G(B) \geq 1$. Another way in which equality can be obtained is when $A = ϕ^{-1}(I), B = ϕ^{-1}(J)$ for some continuous surjective homomorphism $ϕ: G \to {\bf R}/{\bf Z}$ and compact arcs $I,J \subset {\bf R}/{\bf Z}$. We establish an inverse theorem that asserts, roughly speaking, that when equality in the above bound is almost attained, then $A,B$ are close to one of the above examples. We also give a more "robust" form of this theorem in which the sumset $A+B$ is replaced by the partial sumset $A +_\varepsilon B :=\{ 1_A * 1_B \geq \varepsilon \}$ for some small $\varepsilon >0$. In a subsequent paper with Joni Teräväinen, we will apply this latter inverse theorem to establish that certain patterns in multiplicative functions occur with positive density.

math.CO↗

arXiv2017

Some remarks on the lonely runner conjecture

The lonely runner conjecture of Wills and Cusick, in its most popular formulation, asserts that if $n$ runners with distinct constant speeds run around a unit circle ${\bf R}/{\bf Z}$ starting at a common time and place, then each runner will at some time be separated by a distance of at least $\frac{1}{n+1}$ from the others. In this paper we make some remarks on this conjecture. Firstly, we can improve the trivial lower bound of $\frac{1}{2n}$ slightly for large $n$, to $\frac{1}{2n} + \frac{c \log n}{n^2 (\log\log n)^2}$ for some absolute constant $c>0$; previous improvements were roughly of the form $\frac{1}{2n} + \frac{c}{n^2}$. Secondly, we show that to verify the conjecture, it suffices to do so under the assumption that the speeds are integers of size $n^{O(n^2)}$. We also obtain some results in the case when all the velocities are integers of size $O(n)$.

math.CO↗

arXiv2017

Odd order cases of the logarithmically averaged Chowla conjecture

A famous conjecture of Chowla states that the Liouville function $λ(n)$ has negligible correlations with its shifts. Recently, the authors established a weak form of the logarithmically averaged Elliott conjecture on correlations of multiplicative functions, which in turn implied all the odd order cases of the logarithmically averaged Chowla conjecture. In this note, we give a new and shorter proof of the odd order cases of the logarithmically averaged Chowla conjecture. In particular, this proof avoids all mention of ergodic theory, which had an important role in the previous proof.

math.NT↗

arXiv2017

On the universality of the incompressible Euler equation on compact manifolds

The incompressible Euler equations on a compact Riemannian manifold $(M,g)$ take the form \begin{align*} \partial_t u + \nabla_u u &= - \mathrm{grad}_g p \mathrm{div}_g u &= 0. \end{align*} We show that any quadratic ODE $\partial_t y = B(y,y)$, where $B : {\bf R}^n \times {\bf R}^n \to {\bf R}^n$ is a symmetric bilinear map, can be linearly embedded into the incompressible Euler equations for some manifold $M$ if and only if $B$ obeys the cancellation condition $\langle B(y,y), y \rangle = 0$ for some positive definite inner product $\langle,\rangle$ on $ {\bf R}^n$. This allows one to construct explicit solutions to the Euler equations with various dynamical features, such as quasiperiodic solutions, or solutions that transition from one steady state to another, and provides evidence for the "Turing universality" of such Euler flows.

math.AP↗

arXiv2017

New bounds for Szemerédi's theorem, III: A polylogarithmic bound for $r_4(N)$

Define $r_4(N)$ to be the largest cardinality of a set $A \subset \{1,\dots,N\}$ which does not contain four elements in arithmetic progression. In 1998 Gowers proved that \[ r_4(N) \ll N(\log \log N)^{-c}\] for some absolute constant $c>0$. In 2005, the authors improved this to \[ r_4(N) \ll N e^{-c\sqrt{\log\log N}}.\] In this paper we further improve this to \[ r_4(N) \ll N(\log N)^{-c},\] which appears to be the limit of our methods.

math.CO↗

arXiv2017

An integration approach to the Toeplitz square peg problem

The "square peg problem" or "inscribed square problem" of Toeplitz asks if every simple closed curve in the plane inscribes a (non-degenerate) square, in the sense that all four vertices of that square lie on the curve. By a variety of arguments of a "homological" nature, it is known that the answer to this question is positive if the curve is sufficiently regular. The regularity hypotheses are needed to rule out the possibility of arbitrarily small squares that are inscribed or almost inscribed on the curve; because of this, these arguments do not appear to be robust enough to handle arbitrarily rough curves. In this paper we augment the homological approach by introducing certain integrals associated to the curve. This approach is able to give positive answers to the square peg problem in some new cases, for instance if the curve is the union of two Lipschitz graphs $f, g \colon [t_0,t_1] \to \mathbf{R}$ that agree at the endpoints, and whose Lipschitz constants are strictly less than one. We also present some simpler variants of the square problem which seem particularly amenable to this integration approach, including a a periodic version of the problem that is not subject to the problem of arbitrarily small squares (and remains open even for regular curves), as well as an almost purely combinatorial conjecture regarding the sign patterns of sums $y_1+y_2+y_3$ for $y_1,y_2,y_3$ ranging in finite sets of real numbers.

math.GN↗

arXiv2017

A bound on partitioning clusters

Let $X$ be a finite collection of sets (or "clusters"). We consider the problem of counting the number of ways a cluster $A \in X$ can be partitioned into two disjoint clusters $A_1, A_2 \in X$, thus $A = A_1 \uplus A_2$ is the disjoint union of $A_1$ and $A_2$; this problem arises in the run time analysis of the ASTRAL algorithm in phylogenetic reconstruction. We obtain the bound $$ | \{ (A_1,A_2,A) \in X \times X \times X: A = A_1 \uplus A_2 \} | \leq |X|^{3/p} $$ where $|X|$ denotes the cardinality of $X$, and $p := \log_3 \frac{27}{4} = 1.73814\dots$, so that $\frac{3}{p} = 1.72598\dots$. Furthermore, the exponent $p$ cannot be replaced by any larger quantity. This improves upon the trivial bound of $|X|^2$. The argument relies on establishing a one-dimensional convolution inequality that can be established by elementary calculus combined with some numerical verification. In a similar vein, we show that for any subset $A$ of a discrete cube $\{0,1\}^n$, the additive energy of $A$ (the number of quadruples $(a_1,a_2,a_3,a_4)$ in $A^4$ with $a_1+a_2=a_3+a_4$) is at most $|A|^{\log_2 6}$, and that this exponent is best possible.

math.CO↗

arXiv2017

Smooth analysis of the condition number and the least singular value

Let $\a$ be a complex random variable with mean zero and bounded variance. Let $N_{n}$ be the random matrix of size $n$ whose entries are iid copies of $\a$ and $M$ be a fixed matrix of the same size. The goal of this paper is to give a general estimate for the condition number and least singular value of the matrix $M + N_{n}$, generalizing an earlier result of Spielman and Teng for the case when $\a$ is gaussian. Our investigation reveals an interesting fact that the "core" matrix $M$ does play a role on tail bounds for the least singular value of $M+N_{n} $. This does not occur in Spielman-Teng studies when $\a$ is gaussian. Consequently, our general estimate involves the norm $\|M\|$. In the special case when $\|M\|$ is relatively small, this estimate is nearly optimal and extends or refines existing results.

math.PR↗

arXiv2017

An incidence theorem in higher dimensions

We prove almost tight bounds on incidences between points and $k$-dimensional varieties of bounded degree in $\R^d$. Our main tools are the Polynomial Ham Sandwich Theorem and induction on both the dimension and the number of points.

math.CO↗

arXiv2017

Concatenation theorems for anti-Gowers-uniform functions and Host-Kra characteristic factors

We establish a number of "concatenation theorems" that assert, roughly speaking, that if a function exhibits "polynomial" (or "Gowers anti-uniform", "uniformly almost periodic", or "nilsequence") behaviour in two different directions separately, then it also exhibits the same behavior (but at higher degree) in both directions jointly. Among other things, this allows one to control averaged local Gowers uniformity norms by global Gowers uniformity norms. In a sequel to this paper, we will apply such control to obtain asymptotics for "polynomial progressions" $n+P_1(r),\dots,n+P_k(r)$ in various sets of integers, such as the prime numbers.

math.CO↗

arXiv2017

Sum-avoiding sets in groups

Let $A$ be a finite subset of an arbitrary additive group $G$, and let $ϕ(A)$ denote the cardinality of the largest subset $B$ in $A$ that is sum-avoiding in $A$ (that is to say, $b_1+b_2 \not \in A$ for all distinct $b_1,b_2 \in B$). The question of controlling the size of $A$ in terms of $ϕ(A)$ in the case when $G$ was torsion-free was posed by Erdős and Moser. When $G$ has torsion, $A$ can be arbitrarily large for fixed $ϕ(A)$ due to the presence of subgroups. Nevertheless, we provide a qualitative answer to an analogue of the Erdős-Moser problem in this setting, by establishing a structure theorem, which roughly speaking asserts that $A$ is either efficiently covered by $ϕ(A)$ finite subgroups of $G$, or by fewer than $ϕ(A)$ finite subgroups of $G$ together with a residual set of bounded cardinality. In order to avoid a large number of nested inductive arguments, our proof uses the language of nonstandard analysis. We also answer negatively a question of Erdős regarding large subsets $A$ of finite additive groups $G$ with $ϕ(A)$ bounded, but give a positive result when $|G|$ is not divisible by small primes.

math.CO↗

arXiv2017

The Erdos discrepancy problem

We show that for any sequence $f: {\bf N} \to \{-1,+1\}$ taking values in $\{-1,+1\}$, the discrepancy $$ \sup_{n,d \in {\bf N}} \left|\sum_{j=1}^n f(jd)\right| $$ of $f$ is infinite. This answers a question of Erdős. In fact the argument also applies to sequences $f$ taking values in the unit sphere of a real or complex Hilbert space. The argument uses three ingredients. The first is a Fourier-analytic reduction, obtained as part of the Polymath5 project on this problem, which reduces the problem to the case when $f$ is replaced by a (stochastic) completely multiplicative function ${\bf g}$. The second is a logarithmically averaged version of the Elliott conjecture, established recently by the author, which effectively reduces to the case when ${\bf g}$ usually pretends to be a modulated Dirichlet character. The final ingredient is (an extension of) a further argument obtained by the Polymath5 project which shows unbounded discrepancy in this case.

math.CO↗