arXiv subjects

Terence Tao

Publications and source records attributed to Terence Tao.

At least 91 records · Page 5Linked to original sources

arXiv2021

The structure of translational tilings in $\mathbb{Z}^d$

We obtain structural results on translational tilings of periodic functions in $\mathbb{Z}^d$ by finite tiles. In particular, we show that any level one tiling of a periodic set in $\mathbb{Z}^2$ must be weakly periodic (the disjoint union of sets that are individually periodic in one direction), but present a counterexample of a higher level tiling of $\mathbb{Z}^2$ that fails to be weakly periodic. We also establish a quantitative version of the two-dimensional periodic tiling conjecture which asserts that any finite tile in $\mathbb{Z}^2$ that admits a tiling, must admit a periodic tiling, by providing a polynomial bound on the period; this also gives an exponential-type bound on the computational complexity of the problem of deciding whether a given finite subset of $\mathbb{Z}^2$ tiles or not. As a byproduct of our structural theory, we also obtain an explicit formula for a universal period for all tilings of a one-dimensional tile.

math.CA↗

arXiv2021

Zarankiewicz's problem for semilinear hypergraphs

A bipartite graph $H = \left(V_1, V_2; E \right)$ with $|V_1| + |V_2| = n$ is semilinear if $V_i \subseteq \mathbb{R}^{d_i}$ for some $d_i$ and the edge relation $E$ consists of the pairs of points $(x_1, x_2) \in V_1 \times V_2$ satisfying a fixed Boolean combination of $s$ linear equalities and inequalities in $d_1 + d_2$ variables for some $s$. We show that for a fixed $k$, the number of edges in a $K_{k,k}$-free semilinear $H$ is almost linear in $n$, namely $|E| = O_{s,k,\varepsilon}(n^{1+\varepsilon})$ for any $\varepsilon > 0$; and more generally, $|E| = O_{s,k,r,\varepsilon}(n^{r-1 + \varepsilon})$ for a $K_{k, \ldots,k}$-free semilinear $r$-partite $r$-uniform hypergraph. As an application, we obtain the following incidence bound: given $n_1$ points and $n_2$ open boxes with axis parallel sides in $\mathbb{R}^d$ such that their incidence graph is $K_{k,k}$-free, there can be at most $O_{k,\varepsilon}(n^{1+\varepsilon})$ incidences. The same bound holds if instead of boxes one takes polytopes cut out by the translates of an arbitrary fixed finite set of halfspaces. We also obtain matching upper and (superlinear) lower bounds in the case of dyadic boxes on the plane, and point out some connections to the model-theoretic trichotomy in $o$-minimal structures (showing that the failure of an almost linear bound for some definable graph allows one to recover the field operations from that graph in a definable manner).

math.CO↗

arXiv2021

The De Bruijn-Newman constant is non-negative

For each $t \in {\bf R}$, define the entire function $$ H_t(x) := \int_0^\infty e^{tu^2} Φ(u) \cos(xu)\ du$$ where $Φ$ is the super-exponentially decaying function $$ Φ(u) := \sum_{n=1}^\infty (2π^2 n^4 e^{9u} - 3πn^2 e^{5u} ) \exp(-πn^2 e^{4u} ).$$ Newman showed that there exists a finite constant $Λ$ (the \emph{de Bruijn-Newman constant}) such that the zeroes of $H_t$ are all real precisely when $t \geq Λ$. The Riemann hypothesis is the equivalent to the assertion $Λ\leq 0$, and Newman conjectured the complementary bound $Λ\geq 0$. In this paper we establish Newman's conjecture. The argument proceeds by assuming for contradiction that $Λ< 0$, and then analyzing the dynamics of zeroes of $H_t$ (building on the work of Csordas, Smith, and Varga) to obtain increasingly strong control on the zeroes of $H_t$ in the range $Λ< t \leq 0$, until one establishes that the zeroes of $H_0$ are in local equilibrium, in the sense that locally behave (on average) as if they were equally spaced in an arithmetic progression, with gaps staying close to the global average gap size. But this latter claim is inconsistent with the known results about the local distribution of zeroes of the Riemann zeta function, such as the pair correlation estimates of Montgomery.

math.NT↗

arXiv2021

Additive energy of regular measures in one and higher dimensions, and the fractal uncertainty principle

We obtain new bounds on the additive energy of (Ahlfors-David type) regular measures in both one and higher dimensions, which implies expansion results for sums and products of the associated regular sets, as well as more general nonlinear functions of these sets. As a corollary of the higher-dimensional results we obtain some new cases of the fractal uncertainty principle in odd dimensions.

math.CA↗

arXiv2021

Singmaster's conjecture in the interior of Pascal's triangle

Singmaster's conjecture asserts that every natural number greater than one occurs at most a bounded number of times in Pascal's triangle; that is, for any natural number $t \geq 2$, the number of solutions to the equation $\binom{n}{m} = t$ for natural numbers $1 \leq m < n$ is bounded. In this paper we establish this result in the interior region $\exp(\log^{2/3+\varepsilon} n) \leq m \leq n-\exp(\log^{2/3 + \varepsilon} n)$ for any fixed $\varepsilon > 0$. Indeed, when $t$ is sufficiently large depending on $\varepsilon$, we show that there are at most four solutions (or at most two in either half of Pascal's triangle) in this region. We also establish analogous results for the equation $(n)_m = t$, where $(n)_m := n(n-1)\ldots(n-m+1)$ denotes the falling factorial.

math.NT↗

arXiv2021

The Ionescu--Wainger multiplier theorem and the adeles

The Ionescu--Wainger multiplier theorem establishes good $L^p$ bounds for Fourier multiplier operators localized to major arcs; it has become an indispensible tool in discrete harmonic analysis. We give a simplified proof of this theorem with more explicit constants (removing logarithmic losses that were present in previous versions of the theorem), and give a more general variant involving adelic Fourier multipliers. We also establish a closely related adelic sampling theorem that shows that $\ell^p(\Z^d)$ norms of functions with Fourier transform supported on major arcs are comparable to the $L^p(\A_\Z^d)$ norm of their adelic counterparts.

math.CA↗

arXiv2021

The effective potential of an $M$-matrix

In the presence of a confining potential $V$, the eigenfunctions of a continuous Schrödinger operator $-Δ+V$ decay exponentially with the rate governed by the part of $V$ which is above the corresponding eigenvalue; this can be quantified by a method of Agmon. Analogous localization properties can also be established for the eigenvectors of a discrete Schrödinger matrix. This note shows, perhaps surprisingly, that one can replace a discrete Schrödinger matrix by \emph{any} real symmetric $Z$-matrix and still obtain eigenvector localization estimates. In the case of a real symmetric non-singular $M$-matrix $A$ (which is a situation that arises in several contexts, including random matrix theory and statistical physics), the \emph{landscape function} $u = A^{-1} 1$ plays the role of an effective potential of localization. Starting from this potential, one can create an Agmon-type distance function governing the exponential decay of the eigenfunctions away from the "wells" of the potential, a typical eigenfunction being localized to a single such well.

math-ph↗

arXiv2021

Eigenvectors from eigenvalues: A survey of a basic identity in linear algebra

If $A$ is an $n \times n$ Hermitian matrix with eigenvalues $λ_1(A),\dots,λ_n(A)$ and $i,j = 1,\dots,n$, then the $j^{\mathrm{th}}$ component $v_{i,j}$ of a unit eigenvector $v_i$ associated to the eigenvalue $λ_i(A)$ is related to the eigenvalues $λ_1(M_j),\dots,λ_{n-1}(M_j)$ of the minor $M_j$ of $A$ formed by removing the $j^{\mathrm{th}}$ row and column by the formula $$ |v_{i,j}|^2\prod_{k=1;k\neq i}^{n}\left(λ_i(A)-λ_k(A)\right)=\prod_{k=1}^{n-1}\left(λ_i(A)-λ_k(M_j)\right)\,.$$ We refer to this identity as the \emph{eigenvector-eigenvalue identity} and show how this identity can also be used to extract the relative phases between the components of any given eigenvector. Despite the simple nature of this identity and the extremely mature state of development of linear algebra, this identity was not widely known until very recently. In this survey we describe the many times that this identity, or variants thereof, have been discovered and rediscovered in the literature (with the earliest precursor we know of appearing in 1834). We also provide a number of proofs and generalizations of the identity.

math.RA↗

arXiv2020

An arithmetic regularity lemma, associated counting lemma, and applications

Szemeredi's regularity lemma can be viewed as a rough structure theorem for arbitrary dense graphs, decomposing such graphs into a structured piece (a partition into cells with edge densities), a small error (corresponding to irregular cells), and a uniform piece (the pseudorandom deviations from the edge densities). We establish an arithmetic regularity lemma that similarly decomposes bounded functions f : [N] -> C, into a (well-equidistributed, virtual) -step nilsequence, an error which is small in L^2 and a further error which is miniscule in the Gowers U^{s+1}-norm, where s is a positive integer. We then establish a complementary arithmetic counting lemma that counts arithmetic patterns in the nilsequence component of f. We provide a number of applications of these lemmas: a proof of Szemeredi's theorem on arithmetic progressions, a proof of a conjecture of Bergelson, Host and Kra, and a generalisation of certain results of Gowers and Wolf. Our result is dependent on the inverse conjecture for the Gowers U^{s+1} norm, recently established for general s by the authors and T. Ziegler. REVISED November 2020: This paper has been revised so that only systems of linear forms satisfying a condition called the flag property are covered by the counting lemma. Translation-invariant systems, as well as systems of Cauchy-Schwarz complexity 1 and some other naturally-occurring systems have this property. We thank Daniel Altman for drawing our attention to what appeared to be a minor technical issue in one of our proofs but which ultimately led us to realise that the the counting lemma fails quite badly (with rather simple examples, which we shall describe) without some assumption of this type. Additionally, we supply a short appendix showing that the Gowers-Wolf complexity of a system is bounded by its Cauchy-Schwarz complexity.

math.CO↗

arXiv2020

Exploring the toolkit of Jean Bourgain

Gian-Carlo Rota once asserted that "every mathematician only has a few tricks". The sheer breadth and ingenuity in the work of Jean Bourgain may at first glance appear to be a counterexample to this maxim. However, as we hope to illustrate in this article, even Bourgain relied frequently on a core set of tools, which formed the base from which problems in many disparate mathematical fields could then be attacked. We discuss a selected number of these tools here, and then perform a case study of how an argument in one of Bourgain's papers can be interpreted as a sequential application of several of these tools.

math.CA↗

arXiv2020

Quantitative bounds for critically bounded solutions to the Navier-Stokes equations

We revisit the regularity theory of Escauriaza, Seregin, and Šverák for solutions to the three-dimensional Navier-Stokes equations which are uniformly bounded in the critical $L^3_x(\mathbf{R}^3)$ norm. By replacing all invocations of compactness methods in these arguments with quantitative substitutes, and similarly replacing unique continuation and backwards uniqueness estimates by their corresponding Carleman inequalities, we obtain quantitative bounds for higher regularity norms of these solutions in terms of the critical $L^3_x$ bound (with a dependence that is triple exponential in nature). In particular, we show that as one approaches a finite blowup time $T_*$, the critical $L^3_x$ norm must blow up at a rate $(\log\log\log \frac{1}{T_*-t})^c$ or faster for an infinite sequence of times approaching $T_*$ and some absolute constant $c>0$.

math.AP↗

arXiv2020

Homogenization of iterated singular integrals with applications to random quasiconformal maps

We study homogenization of iterated randomized singular integrals and homeomorphic solutions to the Beltrami differential equation with a random Beltrami coefficient. More precisely, let $(F_j)_{j \geq 1}$ be a sequence of normalized homeomorphic solutions to the planar Beltrami equation $\overline{\partial} F_j (z)=μ_j(z,ω) \partial F_j(z),$ where the random dilatation satisfies $|μ_j|\leq k<1$ and has locally periodic statistics, for example of the type $$μ_j (z,ω)=ϕ(z)\sum_{n\in \mathbf{Z}^2}g(2^j z-n,X_{n}(ω)), $$ where $g(z,ω)$ decays rapidly in $z$, the random variables $X_{n}$ are i.i.d., and $ϕ\in C^\infty_0$. We establish the almost sure and local uniform convergence as $j\to\infty$ of the maps $F_j$ to a deterministic quasiconformal limit $F_\infty$. This result is obtained as an application of our main theorem, which deals with homogenization of iterated randomized singular integrals. As a special case of our theorem, let $T_1,\ldots , T_{m}$ be translation and dilation invariant singular integrals on ${\bf R}^d, $ and consider a $d$-dimensional version of $μ_j$, e.g., as defined above or within a more general setting. We then prove that there is a deterministic function $f$ such that almost surely as $j\to\infty$, $$ μ_j T_{m}μ_j\ldots T_1μ_j\to f \quad \textrm{weakly in } L^p,\quad 1 < p < \infty\ . $$

math.CV↗

arXiv2020

Sumset and inverse sumset theorems for Shannon entropy

Let $G = (G,+)$ be an additive group. The sumset theory of Plünnecke and Ruzsa gives several relations between the size of sumsets $A+B$ of finite sets $A, B$, and related objects such as iterated sumsets $kA$ and difference sets $A-B$, while the inverse sumset theory of Freiman, Ruzsa, and others characterises those finite sets $A$ for which $A+A$ is small. In this paper we establish analogous results in which the finite set $A \subset G$ is replaced by a discrete random variable $X$ taking values in $G$, and the cardinality $|A|$ is replaced by the Shannon entropy $\mathrm{Ent}(X)$. In particular, we classify the random variable $X$ which have small doubling in the sense that $\mathrm{Ent}(X_1+X_2) = \mathrm{Ent}(X)+O(1)$ when $X_1,X_2$ are independent copies of $X$, by showing that they factorise as $X = U+Z$ where $U$ is uniformly distributed on a coset progression of bounded rank, and $\mathrm{Ent}(Z) = O(1)$. When $G$ is torsion-free, we also establish the sharp lower bound $\mathrm{Ent}(X+X) \geq \mathrm{Ent}(X) + {1/2} \log 2 - o(1)$, where $o(1)$ goes to zero as $\mathrm{Ent}(X) \to \infty$.

math.CO↗

arXiv2020

On the universality of potential well dynamics

Given a smooth potential function $V : \mathbf{R}^m \to \mathbf{R}$, one can consider the ODE $\partial_t^2 u = -(\nabla V)(u)$ describing the trajectory of a particle $t \mapsto u(t)$ in the potential well $V$. We consider the question of whether the dynamics of this family of ODE are \emph{universal} in the sense that they contain (as embedded copies) any first-order ODE $\partial_t u = X(u)$ arising from a smooth vector field $X$ on a manifold $M$. Assuming that $X$ is nonsingular and $M$ is compact, we show (using the Nash embedding theorem) that this is possible precisely when the flow $(M,X)$ supports a geometric structure which we call a \emph{strongly adapted $1$-form}; many smooth flows do have such a $1$-form, but we give an example (due to Bryant) of a flow which does not, and hence cannot be modeled by the dynamics of a potential well. As one consequence of this embeddability criterion, we construct an example of a (coercive) potential well system which is \emph{Turing complete} in the sense that the halting of any Turing machine with a given input is equivalent to a certain bounded trajectory in this system entering a certain open set. In particular, this system contains trajectories for which it is undecidable whether that trajectory enters such a set. Remarkably, the above results also hold if one works instead with the nonlinear wave equation $\partial_t^2 u - Δu = -(\nabla V)(u)$ on a torus instead of a particle in a potential well, or if one replaces the target domain $\mathbf{R}^m$ by a more general Riemannian manifold.

math.AP↗

arXiv2020

Correlations of the von Mangoldt and higher divisor functions II. Divisor correlations in short ranges

We study the problem of obtaining asymptotic formulas for the sums $\sum_{X < n \leq 2X} d_k(n) d_l(n+h)$ and $\sum_{X < n \leq 2X} Λ(n) d_k(n+h)$, where $Λ$ is the von Mangoldt function, $d_k$ is the $k^{\operatorname{th}}$ divisor function, $X$ is large and $k \geq l \geq 2$ are real numbers. We show that for almost all $h \in [-H, H]$ with $H = (\log X)^{10000 k \log k}$, the expected asymptotic estimate holds. In our previous paper we were able to deal also with the case of $Λ(n) Λ(n + h)$ and we obtained better estimates for the error terms at the price of having to take $H = X^{8/33 + \varepsilon}$.

math.NT↗

arXiv2020

Sharp bounds for multilinear curved Kakeya, restriction and oscillatory integral estimates away from the endpoint

We revisit the multilinear Kakeya, curved Kakeya, restriction, and oscillatory integral estimates that were obtained in paper of Bennett, Carbery, and the author using a heat flow monotonicity method applied to a fractional Cartesian product, together with induction on scales arguments. Many of these estimates contained losses of the form $R^\varepsilon$ (or $\log^{O(1)} R$) for some scale factor $R$. By further developing the heat flow method, and applying it directly for the first time to the multilinear curved Kakeya and restriction settings, we are able to eliminate these losses, as long as the exponent $p$ stays away from the endpoint. In particular, we establish global multilinear restriction estimates away from the endpoint, without any curvature hypotheses on the hypersurfaces.

math.CA↗

arXiv2019

Value patterns of multiplicative functions and related sequences

We study the existence of various sign and value patterns in sequences defined by multiplicative functions or related objects. For any set $A$ whose indicator function is 'approximately multiplicative' and uniformly distributed on short intervals in a suitable sense, we show that the asymptotic density of the pattern $n+1\in A$, $n+2\in A$, $n+3\in A$ is positive, as long as $A$ has density greater than $\frac{1}{3}$. Using an inverse theorem for sumsets and some tools from ergodic theory, we also provide a theorem that deals with the critical case of $A$ having density exactly $\frac{1}{3}$, below which one would need nontrivial information on the local distribution of $A$ in Bohr sets to proceed. We apply our results firstly to answer in a stronger form a question of Erdős and Pomerance on the relative orderings of the largest prime factors $P^{+}(n)$, $P^{+}(n+1), P^{+}(n+2)$ of three consecutive integers. Secondly, we show that the tuple $(ω(n+1),ω(n+2),ω(n+3)) \pmod 3$ takes all the $27$ possible patterns in $(\mathbb{Z}/3\mathbb{Z})^3$ with positive lower density, with $ω(n)$ being the number of distinct prime divisors. We also prove a theorem concerning longer patterns $n+i\in A_i$, $i=1,\dots k$ in approximately multiplicative sets $A_i$ having large enough densities, generalising some results of Hildebrand on his 'stable sets conjecture'. Lastly, we consider the sign patterns of the Liouville function $λ$ and show that there are at least $24$ patterns of length $5$ that occur with positive upper density. In all of the proofs we make extensive use of recent ideas concerning correlations of multiplicative functions.

math.NT↗

arXiv2019

Embedding the Heisenberg group into a bounded dimensional Euclidean space with optimal distortion

Let $H := \begin{pmatrix} 1 & {\mathbf R} & {\mathbf R} \\ 0 & 1 &{\mathbf R} \\ 0 & 0 & 1 \end{pmatrix}$ denote the Heisenberg group with the usual Carnot-Carathéodory metric $d$. It is known (since the work of Pansu and Semmes) that the metric space $(H,d)$ cannot be embedded in a bilipchitz fashion into a Hilbert space; however, from a general theorem of Assouad, for any $0 < \varepsilon < 1$, the snowflaked metric space $(H,d^{1-\varepsilon})$ embeds into an infinite-dimensional Hilbert space with distortion $O( \varepsilon^{-1/2} )$. This distortion bound was shown by Austin, Naor, and Tessera to be sharp for the Heisenberg group $H$. Assouad's argument allows $\ell^2$ to be replaced by ${\mathbf R}^{D(\varepsilon)}$ for some dimension $D(\varepsilon)$ dependent on $\varepsilon$. Naor and Neiman showed that $D$ could be taken independent of $\varepsilon$, at the cost of worsening the bound on the distortion to $O( \varepsilon^{-1+c_D} )$, where $c_D \to 0$ as $D \to \infty$. In this paper we show that one can in fact retain the optimal distortion bound $O( \varepsilon^{-1/2} )$ and still embed into a bounded dimensional space ${\mathbf R}^D$, answering a question of Naor and Neiman. As a corollary, the discrete ball of radius $R \geq 2$ in $Γ:= \begin{pmatrix} 1 & {\mathbf Z} & {\mathbf Z} \\ 0 & 1 & {\mathbf Z} \\ 0 & 0 & 1 \end{pmatrix}$ can be embedded into a bounded-dimensional space ${\mathbf R}^D$ with the optimal distortion bound of $O(\log^{1/2} R)$. The construction is iterative, and is inspired by the Nash-Moser iteration scheme as used in the isometric embedding problem; this scheme is needed in order to counteract a certain ``loss of derivatives'' problem in the iteration.

math.AP↗