Trevor Wooley: Waring’s Problem

30th November, 2022

In 1770, E. Waring made an assertion these days interpreted as conjecturing that when k is a natural number, all positive integers may be written as the sum of a number g(k) of positive integral kth powers, with g(k) finite. Since the work of Hardy and Littlewood a century ago, attention has largely shifted to the problem of bounding G(k), the least number s having the property that all sufficiently large integers can be written as the sum of s positive integral kth powers. It is known that G(2) = 4 (Lagrange), G(3) ≤ 7 (Linnik), G(4) = 16 (Davenport), and G(5) ≤ 17, G(6) ≤ 24, ..., G(20) ≤ 142 (Vaughan and Wooley). For large k one has G(k) ≤ k(log k+log log k+2+o(1)) (Wooley). We report on very recent progress joint with Joerg Bruedern. One or two new world records will be on display.

Daniel Glasscock: Almost all Dynamically Syndetic Sets are Multiplicatively Thick

22nd November, 2022

If a set of integers is syndetic (finitely many translates cover the integers), must it contain two integers whose ratio is a square? No one knows. In the broader context of the disjointness between additive and multiplicative configurations and actions in ergodic Ramsey theory, it makes sense to ask similar questions about dynamically syndetic sets, those sets that contain the visit times of a point to an open set in a minimal topological dynamical system. The main result of the talk is that almost every dynamically syndetic set is multiplicatively very rich: it is 'thick' in some coset of a multiplicative subsemigroup. We will discuss some applications: a 'thick-starters' van der Waerden theorem; the existence of multiplicative structure in sets of the form A – A + t; and the topological disjointness of minimal niltranslations and minimal, aperiodic multiplicative actions.  Time permitting, we will discuss three tools that proved useful in the topic: the prolongation relation (the closure of the orbit-closure relation) developed by Auslander, Akin, and Glasner; the theory of rational points and polynomials on nilmanifolds developed by Leibman, Green, Tao; and the machinery of topological characteristic factors developed recently by Glasner, Huang, Shao, Weiss, and Ye.

Bhavik Mehta: Formalising a proof on unit fractions

15th November, 2022

In late 2021, Bloom resolved an old conjecture of Erdős and Graham, showing that in any subset of the positive integers of positive density, there exists a solution to 1/n1 + 1/n2 + ... + 1/nk = 1 with all ni distinct. We discuss the formal verification of this proof in the Lean theorem prover, including prerequisites from analytic number theory and Fourier analysis, and an instance of the circle method. This is joint work with Thomas Bloom.

James Maynard: Simultaneous Small Fractional Parts of Polynomials

7th November, 2022

Given several real numbers α1,...,αk, how well can you simultaneously approximate all of them by rationals which each have the same square number as a denominator? Schmidt gave a clever iterative argument which showed that this can be done moderately well.

By using a general principle of 'little non-trivial additive structure in rationals' and some ideas from additive combinatorics and the geometry of numbers, I'll describe how this can be improved to give a close-to-optimal answer when k is large.

Thomas Bloom: Sets With Small l1 Fourier Norm

1st November, 2022

A famous conjecture of Littlewood states that the Fourier transform of every set of N integers has l1 norm at least log(N), up to a constant multiplicative factor. This was proved independently by McGehee-Pigno-Smith and Konyagin in the 1980s. This lower bound is the best possible, as it is achieved by an arithmetic progression. An interesting question, especially from the perspective of additive combinatorics, is the 'inverse problem': what can we say about sets which are close to optimal, say with l1 norm at most 100 log(N)? I will discuss can inverse result of this type, showing that (in a certain sense) such sets are approximately the union of O(1) sets with small doubling.

Izabella Łaba: Cyclotomic divisibility – from tiling to harmonic analysis and geometric measure theory

30th March, 2022

It is well known that if a finite set of integers A tiles the integers by translations, then the translation set must be periodic, so that the tiling is equivalent to a factorization A+B=ZM of a finite cyclic group. Coven and Meyerowitz (1998) proposed a characterization of all finite tiles in terms of the cyclotomic divisors of associated mask polynomials, and proved it when the tiling period M has at most two distinct prime factors. In joint work with Itay Londner, we extended it to the case when M=(pqr)2, where p,q,r are distinct primes. The methods we developed can be applied to other questions that hinge on cyclotomic divisibility, ranging from number theory to harmonic analysis and geometric measure theory. In particular, Caleb Marshall and I were able to use cyclotomic divisibility methods to prove new Favard length estimates for product Cantor sets. The talk will provide an introduction to this group of problems.

Theresa Anderson: Two meetings of number theory and analysis

16th March, 2022

In many recent works, analysis and number theory go beyond working side by side and team up in an interconnected back and forth interplay to become a powerful force. Here I describe two distinct meetings of the pair, which result in sharp counts for equilateral triangles in Euclidean space and statistics for how often a random polynomial has Galois group not isomorphic to the full symmetric group.

Matthew Tointon: Percolation on finite transitive graphs

2nd March, 2022

In Bernoulli bond percolation, one defines a random subgraph of a given connected graph G by deleting or retaining edges of G independently at random, each edge being retained with the same probability p. When G is infinite, a central and classical question is whether there exists a choice of p strictly less than 1 such that this random subgraph has infinite connected components. When G is finite, a natural analogue is to ask whether there exists a choice of p bounded away from 1 such that the random subgraph contains a connected component containing at least half (or 1% or 99%) of the vertices of G. Tom Hutchcroft and I recently showed that such a p exists provided G is not close to a cycle in some sense, confirming most cases of a conjecture of Benjamini (2001). I will give an overview of the proof, and go into some detail on certain aspects of it that have a distinctly additive combinatorial flavour.

Max Wenqiang Xu: Product sets of arithmetic progressions

16th February, 2022

We prove that the size of the product set of any finite arithmetic progression A in integers of size N is at least N2/(log N)c+o(1), where c=1-(1+loglog 2)/(log 2). This matches the bound in the celebrated Erdos multiplication table problem, up to a factor of (log N)o(1) and thus confirms a conjecture of Elekes and Ruzsa. If instead A is relaxed to be a subset of a finite arithmetic progression in integers with positive constant density, we prove that the size of the product set is at least N2/(log N)2log2-1 + o(1). This solves the typical case of another conjecture of Elekes and Ruzsa on the size of the product set of a set A whose sum set is of size O(|A|).This is joint work with Yunkun Zhou.

Hunter Spink: Anti-concentration of random walks on model-theoretic definable sets

2nd February, 2022

(Joint with Jacob Fox, Matthew Kwan) Classical anti-concentration results show "random walks in ℝd with BIG independent steps can't concentrate in balls much better than they can concentrate on individual points". Model-theoretic definable sets include boolean combinations of subsets of ℝd defined using equalities and inequalities of arbitrary compositions of polynomials, ex, ln(x) and analytic functions restricted to compact boxes. For example, the intersection of esin(1/(1+(xyz)2))+x2y+zy ≥ 0 and xyz=5 in ℝ3. In this talk, I will discuss recent results which show "random walks in ℝd with ARBITRARY independent steps can't concentrate in definable sets not containing line segments much better than they can concentrate on individual points". Time permitting, I will discuss how these results extend to other groups like GLd(ℝ).