65 Result(s)
within
Article
Terence Tao
-
Article
Open AccessMonotone Nondecreasing Sequences of the Euler Totient Function
Let M(x) denote the largest cardinality of a subset of \(\{n \in \mathbb {N}: n \le x\}\) ...
-
Article
Open AccessPerfectly Packing a Square by Squares of Nearly Harmonic Sidelength
A well-known open problem of Meir and Moser asks if the squares of sidelength 1/n for \(n\ge 2\) ...
-
Article
Sums of GUE matrices and concentration of hives from correlation decay of eigengaps
Associated to two given sequences of eigenvalues \(\lambda _1 \ge \cdots \ge \lambda _n\) ...
-
Article
Infinite partial sumsets in the primes
We show that there exist infinite sets A = (a1, a2, …} and B = {b1, b2, …} of natural numbers such that ai + bj is prime whenever 1 ≤ i < j.
-
Article
Open AccessUndecidable Translational Tilings with Only Two Tiles, or One Nonabelian Tile
We construct an example of a group \(G = \mathbb {Z}^2 \times G_0\) G ...
-
Article
Open AccessLarge prime gaps and probabilistic models
We introduce a new probabilistic model of the primes consisting of integers that survive the sieving process when a random residue class is selected for every prime modulus below a specific bound. From a rigor...
-
Article
Optimal Sine and Sawtooth Inequalities
We determine the optimal inequality of the form \(\sum _{k=1}^m a_k\sin kx\le 1\) ...
-
Article
Correction to: Fourier uniformity of bounded multiplicative functions in short intervals on average
The original version of this article unfortunately contains a mistake.
-
Article
Fourier uniformity of bounded multiplicative functions in short intervals on average
Let \(\lambda \)λ denote the Liouville function. We show that as \(X \rightarrow \infty \)X→∞, $$\begin{aligned} \int _{X}^{2X} \sup _{\alpha } \left| \sum _{x < n \le x + H} \lambda (n) e(-\alpha n) \right| dx =...
-
Article
Searching for singularities in the Navier–Stokes equations
Despite much effort, the question of whether the Navier–Stokes equations allow solutions that develop singularities in finite time remains unresolved. Terence Tao discusses the problem, and possible routes to ...
-
Article
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 \(\sum _{X < n \le 2X} d_k(n) d_l(n+h)\) ...
-
Article
An Inverse Theorem for an Inequality of Kneser
Let G = (G, +) be a compact connected abelian group, and let μG denote its probability Haar measure. A theorem of Kneser (generalising previous results of Macbeath, Raikov, and Shields) establishes the bound μG(A
-
Article
Random matrices have simple spectrum
Let M n =(ξ ij )1≤i,j≤n be a real symmetric ...
-
Article
Random matrices: tail bounds for gaps between eigenvalues
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 t...
-
Article
Finite Time Blowup for Lagrangian Modifications of the Three-Dimensional Euler Equation
In the language of differential geometry, the incompressible inviscid Euler equations can be written in vorticity-vector potential form as ...
-
Article
Cancellation for the multilinear Hilbert transform
For any natural number k, consider the k-linear Hilbert transform $$\begin{aligned} H_k( f_1,\dots ,f_k )(x) := {\text {p.v.}} \int _\...
-
Article
Multiple recurrence and convergence results associated to \({\text{F}}_P^\omega \) -actions
Using an ergodic inverse theorem obtained in our previous paper, we obtain limit formulae for multiple ergodic averages associated with the action of ...
-
Article
A multi-dimensional Szemerédi theorem for the primes via a correspondence principle
We establish a version of the Furstenberg-Katznelson multi-dimensional Szemerédi theorem in the primes P:= {2, 3, 5, …}, which roughly speaking asserts that any dense subset of P ...
-
Article
Effective Limiting Absorption Principles, and Applications
The limiting absorption principle asserts that if H is a suitable Schrödinger operator, and f lives in a suitable weighted L 2 space (namely \({H^{0, 1/2 + \sigma}}\) ...
-
Article
Multiple Recurrence in Quasirandom Groups
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, th...
