Blog Profile / Terence Tao's Blog

Filed Under:Academics / Mathematics
Posts on Regator:384
Posts / Week:1.1
Archived Since:December 12, 2008

Blog Post Archive

Nested approximate subgroups

Suppose that are two subgroups of some ambient group, with the index of in being finite. Then is the union of left cosets of, thus for some set of cardinality. The elements of are not entirely arbitrary with regards to. For instance, if is a normal subgroup of, then for […]

A differentiation identity

Here’s a cute identity I discovered by accident recently. Observe that and so one can conjecture that one has when is even, and when is odd. This is obvious in the even case since is a polynomial of degree, but I struggled for a while with the odd case before finding a slick three-line […]

Cancellation for the multilinear Hilbert transform

I’ve just uploaded to the arXiv my paper “Cancellation for the multilinear Hilbert transform“, submitted to Collectanea Mathematica. This paper uses methods from additive combinatorics (and more specifically, the arithmetic regularity and counting lemmas from this paper of Ben Green and myself) to obtain a slight amount of progress towards the open problem of obtaining […]

Failure of the L^1 pointwise and maximal ergodic theorems for the free group

I’ve just uploaded to the arXiv my paper “Failure of the pointwise and maximal ergodic theorems for the free group“, submitted to Forum of Mathematics, Sigma. This paper concerns a variant of the pointwise ergodic theorem of Birkhoff, which asserts that if one has a measure-preserving shift map on a probability space, then for […]

A remark on the lonely runner conjecture

The lonely runner conjecture is the following open problem: Conjecture 1 Suppose one has runners on the unit circle, all starting at the origin and moving at different speeds. Then for each runner, there is at least one time for which that runner is “lonely” in the sense that it is separated by a […]

The standard branch of the matrix logarithm

Because of Euler’s identity, the complex exponential is not injective: for any complex and integer. As such, the complex logarithm is not well-defined as a single-valued function from to. However, after making a branch cut, one can create a branch of the logarithm which is single-valued. For instance, after removing the negative […]

Embedding the SQG equation in a modified Euler equation

The Euler equations for three-dimensional incompressible inviscid fluid flow are where is the velocity field, and is the pressure field. For the purposes of this post, we will ignore all issues of decay or regularity of the fields in question, assuming that they are as smooth and rapidly decreasing as needed to justify all the […]

Newton iteration and the Siegel linearisation theorem

An extremely large portion of mathematics is concerned with locating solutions to equations such as or for in some suitable domain space (either finite-dimensional or infinite-dimensional), and various maps or. To solve the fixed point iteration equation (1), the simplest general method available is the fixed point iteration method: one starts with an initial […]

The ergodic theorem and Gowers-Host-Kra seminorms without separability or amenability

The von Neumann ergodic theorem (the Hilbert space version of the mean ergodic theorem) asserts that if is a unitary operator on a Hilbert space, and is a vector in that Hilbert space, then one has in the strong topology, where is the -invariant subspace of, and is the orthogonal projection to. […]

Random matrices: tail bounds for gaps between eigenvalues

Hoi Nguyen, Van Vu, and myself have just uploaded to the arXiv our paper “Random matrices: tail bounds for gaps between eigenvalues“. This is a followup paper to my recent paper with Van in which we showed that random matrices of Wigner type (such as the adjacency graph of an Erd\H{o}s-Renyi graph) almost surely had […]

254A, Notes 8: The Hardy-Littlewood circle method and Vinogradov’s theorem

We have seen in previous notes that the operation of forming a Dirichlet series or twisted Dirichlet series is an incredibly useful tool for questions in multiplicative number theory. Such series can be viewed as a multiplicative Fourier transform, since the functions and are multiplicative characters. Similarly, it turns out that the operation of forming […]

An averaged form of Chowla’s conjecture

Kaisa Matomaki, Maksym Radziwill, and I have just uploaded to the arXiv our paper “An averaged form of Chowla’s conjecture“. This paper concerns a weaker variant of the famous conjecture of Chowla (discussed for instance in this previous post) that as for any distinct natural numbers, where denotes the Liouville function. (One could also […]

254A, Supplement 7: Normalised limit profiles of the log-magnitude of the Riemann zeta function (optional)

A major topic of interest of analytic number theory is the asymptotic behaviour of the Riemann zeta function in the critical strip in the limit. For the purposes of this set of notes, it is a little simpler technically to work with the log-magnitude of the zeta function. (In principle, one can reconstruct a […]

254A, Supplement 6: A cheap version of the theorems of Halasz and Matomaki-Radziwill (optional)

In analytic number theory, it is a well-known phenomenon that for many arithmetic functions of interest in number theory, it is significantly easier to estimate logarithmic sums such as than it is to estimate summatory functions such as (Here we are normalising to be roughly constant in size, e.g. as.) For instance, when is […]

254A, Notes 7: Linnik’s theorem on primes in arithmetic progressions

In the previous set of notes, we saw how zero-density theorems for the Riemann zeta function, when combined with the zero-free region of Vinogradov and Korobov, could be used to obtain prime number theorems in short intervals. It turns out that a more sophisticated version of this type of argument also works to obtain prime […]

254A, Notes 6: Large values of Dirichlet polynomials, zero density estimates, and primes in short intervals

In the previous set of notes, we studied upper bounds on sums such as for that were valid for all in a given range, such as ; this led in turn to upper bounds on the Riemann zeta for in the same range, and for various choices of. While some improvement over the trivial […]

254A, Notes 5: Bounding exponential sums and the zeta function

We return to the study of the Riemann zeta function, focusing now on the task of upper bounding the size of this function within the critical strip; as seen in Exercise 43 of Notes 2, such upper bounds can lead to zero-free regions for, which in turn lead to improved estimates for the […]

254A, Supplement 5: The linear sieve and Chen’s theorem (optional)

We continue the discussion of sieve theory from Notes 4, but now specialise to the case of the linear sieve in which the sieve dimension is equal to, which is one of the best understood sieving situations, and one of the rare cases in which the precise limits of the sieve method are known. […]

254A, Notes 4: Some sieve theory

Many problems in non-multiplicative prime number theory can be recast as sieving problems. Consider for instance the problem of counting the number of pairs of twin primes contained in for some large ; note that the claim that for arbitrarily large is equivalent to the twin prime conjecture. One can obtain this count by any […]

254A, Notes 3: The large sieve and the Bombieri-Vinogradov theorem

A fundamental and recurring problem in analytic number theory is to demonstrate the presence of cancellation in an oscillating sum, a typical example of which might be a correlation between two arithmetic functions and, which to avoid technicalities we will assume to be finitely supported (or that the variable is localised to a finite […]

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