arXiv subjects

Terence Tao

Publications and source records attributed to Terence Tao.

At least 55 records · Page 3Linked to original sources

arXiv2024

A counterexample to the periodic tiling conjecture

The periodic tiling conjecture asserts that any finite subset of a lattice $\mathbb{Z}^d$ which tiles that lattice by translations, in fact tiles periodically. In this work we disprove this conjecture for sufficiently large $d$, which also implies a disproof of the corresponding conjecture for Euclidean spaces $\mathbb{R}^d$. In fact, we also obtain a counterexample in a group of the form $\mathbb{Z}^2 \times G_0$ for some finite abelian $2$-group $G_0$. Our methods rely on encoding a "Sudoku puzzle" whose rows and other non-horizontal lines are constrained to lie in a certain class of "$2$-adically structured functions," in terms of certain functional equations that can be encoded in turn as a single tiling equation, and then demonstrating that solutions to this Sudoku puzzle exist, but are all non-periodic.

math.CO↗

arXiv2024

Planar point sets with forbidden $4$-point patterns and few distinct distances

We show that for any large $n$, there exists a set of $n$ points in the plane with $O(n^2/\sqrt{\log n})$ distinct distances, such that any four points in the set determine at least five distinct distances. This answers (in the negative) a question of Erdős. The proof combines an analysis by Dumitrescu of forbidden four-point patterns with an algebraic construction of Thiele and Dumitrescu (to eliminate parallelograms), as well as a randomized transformation of that construction (to eliminate most other forbidden patterns).

math.CO↗

arXiv2024

Maximal multilinear operators

We establish multilinear $L^p$ bounds for a class of maximal multilinear averages of functions on one variable, reproving and generalizing the bilinear maximal function bounds of Lacey. As an application we obtain almost everywhere convergence results for these averages, and in some cases we also obtain almost everywhere convergence for their ergodic counterparts on a dynamical system.

math.CA↗

arXiv2024

Random matrices: Universality of local spectral statistics of non-Hermitian matrices

It is a 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 $\mathbb{C}$ with kernel $K_{\infty}(z,w):=\frac{1}πe^{-|z|^2/2-|w|^2/2+z\bar{w}}$ in the limit $n\to\infty$. 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. 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. These are non-Hermitian analogues of some recent universality results for Hermitian Wigner matrices. However, a new difficulty arises in the non-Hermitian case, due to the instability of the spectrum for such matrices. To resolve this, we work with the log-determinants $\log|\det(M_n-z_0)|$ rather than with the Stieltjes transform $\frac{1}{n}\operatorname {tr}(M_n-z_0)^{-1}$. Our main tools are a four moment theorem for these log-determinants, together with a strong concentration result for the log-determinants in the Gaussian case. The latter is established by studying the solutions of a certain nonlinear stochastic difference equation. With some extra consideration, we can extend our arguments to the real case, proving universality for correlation functions of real matrices which match the real Gaussian ensemble to the fourth order. 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 $\sqrt{\frac{2n}π}+o(\sqrt{n})$ real eigenvalues asymptotically almost surely.

math.PR↗

arXiv2024

A lower bound on the mean value of the Erdős-Hooley Delta function

We give an improved lower bound for the average of the Erdős-Hooley function $Δ(n)$, namely $\sum_{n\le x} Δ(n) \gg_\varepsilon x(\log\log x)^{1+η-\varepsilon}$ for all $x\geqslant100$ and any fixed $\varepsilon$, where $η= 0.3533227\dots$ is an exponent previously appearing in work of Green and the first two authors. This improves on a previous lower bound of $\gg x \log\log x$ of Hall and Tenenbaum, and can be compared to the recent upper bound of $x (\log\log x)^{11/4}$ of the second and third authors.

math.NT↗

arXiv2024

Marton's Conjecture in abelian groups with bounded torsion

We prove a Freiman--Ruzsa-type theorem with polynomial bounds in arbitrary abelian groups with bounded torsion, thereby proving (in full generality) a conjecture of Marton. Specifically, let $G$ be an abelian group of torsion $m$ (meaning $mg=0$ for all $g \in G$) and suppose that $A$ is a non-empty subset of $G$ with $|A+A| \leq K|A|$. Then $A$ can be covered by at most $(2K)^{O(m^3)}$ translates of a subgroup of $H \leq G$ of cardinality at most $|A|$. The argument is a variant of that used in the case $G = \mathbf{F}_2^n$ in a recent paper of the authors.

math.NT↗

arXiv2024

Monotone non-decreasing sequences of the Euler totient function

Let $M(x)$ denote the largest cardinality of a subset of $\{n \in \mathbf{N}: n \leq x\}$ on which the Euler totient function $φ(n)$ is non-decreasing. We show that $M(x) = (1+O(\frac{(\log\log x)^5}{\log x})) π(x)$ for all $x \geq 10$, answering questions of Erdős and Pollack--Pomerance--Treviño. A similar result is also obtained for the sum of divisors function $σ(n)$.

math.NT↗

arXiv2024

Higher uniformity of arithmetic functions in short intervals I. All intervals

We study higher uniformity properties of the Möbius function $μ$, the von Mangoldt function $Λ$, and the divisor functions $d_k$ on short intervals $(X,X+H]$ with $X^{θ+\varepsilon} \leq H \leq X^{1-\varepsilon}$ for a fixed constant $0 \leq θ< 1$ and any $\varepsilon>0$. More precisely, letting $Λ^\sharp$ and $d_k^\sharp$ be suitable approximants of $Λ$ and $d_k$ and $μ^\sharp = 0$, we show for instance that, for any nilsequence $F(g(n)Γ)$, we have \[ \sum_{X < n \leq X+H} (f(n)-f^\sharp(n)) F(g(n) Γ) \ll H \log^{-A} X \] when $θ= 5/8$ and $f \in \{Λ, μ, d_k\}$ or $θ= 1/3$ and $f = d_2$. As a consequence, we show that the short interval Gowers norms $\|f-f^\sharp\|_{U^s(X,X+H]}$ are also asymptotically small for any fixed $s$ for these choices of $f,θ$. As applications, we prove an asymptotic formula for the number of solutions to linear equations in primes in short intervals, and show that multiple ergodic averages along primes in short intervals converge in $L^2$. Our innovations include the use of multi-parameter nilsequence equidistribution theorems to control type $II$ sums, and an elementary decomposition of the neighbourhood of a hyperbola into arithmetic progressions to control type $I_2$ sums.

math.NT↗

arXiv2024

The structure of arbitrary Conze-Lesigne systems

Let $Γ$ be a countable abelian group. An (abstract) $Γ$-system $\mathrm{X}$ - that is, an (abstract) probability space equipped with an (abstract) probability-preserving action of $Γ$ - is said to be a Conze-Lesigne system if it is equal to its second Host-Kra-Ziegler factor $\mathrm{Z}^2(\mathrm{X})$. The main result of this paper is a structural description of such Conze-Lesigne systems for arbitrary countable abelian $Γ$, namely that they are the inverse limit of translational systems $G_n/Λ_n$ arising from locally compact nilpotent groups $G_n$ of nilpotency class $2$, quotiented by a lattice $Λ_n$. Results of this type were previously known when $Γ$ was finitely generated, or the product of cyclic groups of prime order. In a companion paper, two of us will apply this structure theorem to obtain an inverse theorem for the Gowers $U^3(G)$ norm for arbitrary finite abelian groups $G$.

math.DS↗

arXiv2024

Undecidable translational tilings with only two tiles, or one nonabelian tile

We construct an example of a group $G = \mathbb{Z}^2 \times G_0$ for a finite abelian group $G_0$, a subset $E$ of $G_0$, and two finite subsets $F_1,F_2$ of $G$, such that it is undecidable in ZFC whether $\mathbb{Z}^2\times E$ can be tiled by translations of $F_1,F_2$. In particular, this implies that this tiling problem is aperiodic, in the sense that (in the standard universe of ZFC) there exist translational tilings of $E$ by the tiles $F_1,F_2$, but no periodic tilings. Previously, such aperiodic or undecidable translational tilings were only constructed for sets of eleven or more tiles (mostly in $\mathbb{Z}^2$). A similar construction also applies for $G = \mathbb{Z}^d$ for sufficiently large $d$. If one allows the group $G_0$ to be non-abelian, a variant of the construction produces an undecidable translational tiling with only one tile $F$. The argument proceeds by first observing that a single tiling equation is able to encode an arbitrary system of tiling equations, which in turn can encode an arbitrary system of certain functional equations once one has two or more tiles. In particular, one can use two tiles to encode tiling problems for an arbitrary number of tiles.

math.CO↗

arXiv2024

Infinite partial sumsets in the primes

We show that there exist infinite sets $A = \{a_1,a_2,\dots\}$ and $B = \{b_1,b_2,\dots\}$ of natural numbers such that $a_i+b_j$ is prime whenever $1 \leq i < j$.

math.NT↗

arXiv2024

New bounds for Szemeredi's theorem, II: A new bound for $r_4(N)$

Define $r_4(N)$ to be the largest cardinality of a set $A$ in $\{1,\dots,N\}$ 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 this paper (part II of a series) we improve this to $r_4(N) \ll N e^{-c\sqrt{\log \log N}}$. In part III of the series we will use a more elaborate argument to improve this to $r_4(N) \ll N(\log N)^{-c}$.

math.NT↗

arXiv2023

On a conjecture of Marton

We prove a conjecture of K. Marton, widely known as the polynomial Freiman--Ruzsa conjecture, in characteristic $2$. The argument extends to odd characteristic, with details to follow in a subsequent paper.

math.NT↗

arXiv2023

An inverse theorem for the Gowers U^3 norm

The Gowers U^3 norm is one of a sequence of norms used in the study of arithmetic progressions. If G is an abelian group and A is a subset of G then the U^3(G) of the characteristic function 1_A is useful in the study of progressions of length 4 in A. We give a comprehensive study of the U^3(G) norm, obtaining a reasonably complete description of functions f : G -> C for which ||f||_{U^3} is large and providing links to recent results of Host, Kra and Ziegler in ergodic theory. As an application we generalise a result of Gowers on Szemeredi's theorem. Writing r_4(G) for the size of the largest set A not containing four distinct elements in arithmetic progression, we show that r_4(G) << |G|(loglog|G|)^{-c} for some absolute constant c. In future papers we will develop these ideas further, obtaining an asymptotic for the number of 4-term progressions p_1 < p_2 < p_3 < p_4 < N of primes as well as superior bounds for r_4(G). Update, December 2023. Proposition 3.2 in the paper, which is stated without detailed proof, is incorrect. For a counterexample, see Candela, Gonzalez-Sanchez and Szegedy arXiv:2311.13899, Remark 4.3. Proposition 3.2 is invoked twice in the paper. First, it is used immediately after its statement to deduce the second part of Theorem 2.3. However, that theorem concerns only vector spaces over finite fields, and in this setting Proposition 3.2 is correct by standard linear algebra. The remark at the end of Section 3 that the argument works for arbitrary $G$ should, however, be deleted. The second application is in the proof of Lemma 10.6. It may well be possible to salvage this lemma, particularly if $P$ is assumed proper, but in any case it is only applied once, in the proof of Proposition 10.8. There, $P$ is proper and, more importantly, $H = \{0\}$ is trivial; in this setting Lemma 10.6 and its proof remain valid.

math.NT↗

arXiv2023

A Maclaurin type inequality

The classical Maclaurin inequality asserts that the elementary symmetric means $$ s_k(y) = \frac{1}{\binom{n}{k}} \sum_{1 \leq i_1 < \dots < i_k \leq n} y_{i_1} \dots y_{i_k}$$ obey the inequality $s_\ell(y)^{1/\ell} \leq s_k(y)^{1/k}$ whenever $1 \leq k \leq \ell \leq n$ and $y = (y_1,\dots,y_n)$ consists of non-negative reals. We establish a variant $$ |s_\ell(y)|^{\frac{1}{\ell}} \ll \frac{\ell^{1/2}}{k^{1/2}} \max (|s_k(y)|^{\frac{1}{k}}, |s_{k+1}(y)|^{\frac{1}{k+1}})$$ of this inequality in which the $y_i$ are permitted to be negative. In this regime the inequality is sharp up to constants. Such an inequality was previously known without the $k^{1/2}$ factor in the denominator.

math.CA↗

arXiv2023

An upper bound on the mean value of the Erdős-Hooley Delta function

The Erdős-Hooley Delta function is defined for $n\in\mathbb{N}$ as $Δ(n)=\sup_{u\in\mathbb{R}} \#\{d|n : e^u<d\le e^{u+1}\}$. We prove that $\sum_{n\le x} Δ(n) \ll x(\log\log x)^{11/4}$ for all $x\ge100$. This improves on earlier work of Hooley, Hall--Tenenbaum and La Bretèche-Tenenbaum.

math.NT↗

arXiv2023

Quantitative bounds for Gowers uniformity of the Möbius and von Mangoldt functions

We establish quantitative bounds on the $U^k[N]$ Gowers norms of the Möbius function $μ$ and the von Mangoldt function $Λ$ for all $k$, with error terms of shape $O((\log\log N)^{-c})$. As a consequence, we obtain quantitative bounds for the number of solutions to any linear system of equations of finite complexity in the primes, with the same shape of error terms. We also obtain the first quantitative bounds on the size of sets containing no $k$-term arithmetic progressions with shifted prime difference.

math.NT↗

arXiv2023

Undecidability of translational monotilings

In the 60's, Berger famously showed that translational tilings of $\mathbb{Z}^2$ with multiple tiles are algorithmically undecidable. Recently, Bhattacharya proved the decidability of translational monotilings (tilings by translations of a single tile) in $\mathbb{Z}^2$. The decidability of translational monotilings in higher dimensions remained unsolved. In this paper, by combining our recently developed techniques with ideas introduced by Aanderaa and Lewis, we finally settle this problem, achieving the undecidability of translational monotilings of (periodic subsets of) virtually $\mathbb{Z}^2$ spaces, namely, spaces of the form $\mathbb{Z}^2\times G_0$, where $G_0$ is a finite Abelian group. This also implies the undecidability of translational monotilings in $\mathbb{Z}^d$, $d\geq 3$.

math.CO↗