We prove that any set $A\subset \mathbb{N}$ of positive upper density contains a finite $S\subset A$ such that $\sum_{n\in S}\frac{1}{n}=1$, answering a question of Erdős and Graham.
We show that if $A\subset \{1,\ldots,N\}$ has no solutions to $a-b=n^2$ with $a,b\in A$ and $n\geq 1$ then \[ \lvert A\rvert \ll \frac{N}{(\log N)^{c\log\log\log N}}\] for some absolute constant $c>0$. This improves upon a result of Pintz-Steiger-Szemerédi.
(This text is a survey written for the Bourbaki seminar on the work of F. Manners.)
Gowers uniformity norms are the central objects of higher order Fourier analysis, one of the cornerstones of additive combinatorics, and play an important role in both Gowers' proof of Szemerédi's theorem and the Green-Tao theorem. The inverse theorem states that if a function has a large uniformity norm, which is a robust combinatorial measure of structure, then it must correlate with a nilsequence, which is a highly structured algebraic object. This was proved in a qualitative sense by Green, Tao, and Ziegler, but with a proof that was incapable of providing reasonable bounds. In 2018 Manners achieved a breakthrough by giving a new proof of the inverse theorem. Not only does this new proof contain a wealth of new insights but it also, for the first time, provides quantitative bounds, that are at worst only doubly exponential. This talk will give a high-level overview of what the inverse theorem says, why it is important, and the new proof of Manners.
We show that if $A\subset \{1,\ldots,N\}$ contains no non-trivial three-term arithmetic progressions then $\lvert A\rvert \ll N/(\log N)^{1+c}$ for some absolute constant $c>0$. In particular, this proves the first non-trivial case of a conjecture of Erdős on arithmetic progressions.
We give a new proof of logarithmic bounds for Roth's theorem on arithmetic progressions, namely that if $A\subset \{1,2,\ldots,N\}$ is free of three-term progressions, then $\lvert A\rvert \leq N/(\log N)^{1-o(1)}$. Unlike previous proofs, this is almost entirely done in physical space using almost-periodicity.
In this note we prove a new estimate on so-called GCD sums (also called Gál sums), which, for certain coefficients, improves significantly over the general bound due to de la Bretêche and Tenenbaum. We use our estimate to prove new results on the equidistribution of sequences modulo 1, improving over a result of Aistleitner, Larcher, and Lewko on how the metric poissonian property relates to the notion of additive energy. In particular, we show that arbitrary subsets of the squares are metric poissonian.
Roughly, a set $A$ of natural numbers is metric Poissonian if the dilates of differences $\alpha(a-b)$ for $\alpha \in (0,1)$ and $a,b\in A$ are on average well-distributed. We show that if $A$ has small additive energy, i.e. relatively few solutions to $a-b=c-d$, and is relatively dense, then $A$ is metric Poissonian.
In which we show that for \(n\geq 4\) the graphs \(K_n\) and \(K_n+K_{n-1}\) are Ramsey equivalent; that is, if a graph \(G\) has the property that, given any red-blue colouring of the edges of \(G\), a monochromatic copy of \(K_n\) necessarily appears somewhere, then a monochromatic copy of \(K_n+K_{n-1}\) is also forced.
I obtain an improved quantitative version of Roth's theorem, showing that if \(A\subset \{1,\ldots,N\}\) contains no non-trivial three-term arithmetic progressions then \(\lvert A\rvert \ll N(\log\log N)^4/\log N\). The method used is quite different to that used by Sanders, who earlier obtained a similar bound, and is based on the techniques used by Bateman and Katz in their work on the analogous problem over \(\mathbb{F}_3^n\).
We show that, for any finite set \(A\) which lives in either a rational function field \(\mathbb{F}_q(t)\) or a p-adic field \(\mathbb{Q}_p\), either the sum set \(A+A\) or the product set \(A\cdot A\) has cardinality at least \(\lvert A\rvert^{6/5-o(1)}\).
I extend the improvement of Roth's theorem on three term arithmetic progressions by Sanders to obtain similar results for the problem of locating non-trivial solutions to translation-invariant linear equations in many variables (e.g. \(x_1+x_2+x_3=3x_4\)) in both \(\mathbb{Z}/N\mathbb{Z}\) and \(\mathbb{F}_q[t]\).