In this expository note, we give a concise and accessible introduction to the real-analytic determinant method for counting integral points on algebraic curves, based on the classic 1989 paper of Bombieri-Pila.

We prove near-optimal upper bounds for the odd moments of the distribution of coprime residues in short intervals, confirming a conjecture of Montgomery and Vaughan. As an application we prove near-optimal upper bounds for the average of the refined singular series in the Hardy-Littlewood conjectures concerning the number of prime $k$-tuples for $k$ odd. The main new ingredient is a near-optimal upper bound for the number of solutions to $\sum_{1\leq i\leq k}\frac{a_i}{q_i}\in\mathbb{Z}$ when $k$ is odd, with $(a_i,q_i)=1$ and restrictions on the size of the numerators and denominators, that is of independent interest.

In a recent breakthrough Kelley and Meka proved a quasipolynomial upper bound for the density of sets of integers without non-trivial three-term arithmetic progressions. We present a simple modification to their method that strengthens their conclusion, in particular proving that if $A\subseteq \{1,\ldots,N\}$ has no non-trivial three-term arithmetic progressions then $\lvert A\rvert \leq \exp(-c(\log N)^{1/9})N$ for some $c > 0$.

We prove that any positive rational number is the sum of distinct unit fractions with denominators in $\{p-1 : p \textrm{ prime}\}$. The same conclusion holds for the set $\{ p-h : p \textrm{ prime}\}$ for any $h\in\mathbb{Z}\backslash \{0\}$, provided a necessary congruence condition is satisfied. We also prove that this is true for any subset of the primes of relative positive density, provided a necessary congruence condition is satisfied.

We give a self-contained exposition of the recent remarkable result of Kelley and Meka: if $A\subseteq \{1,\ldots,N\}$ has no non-trivial three-term arithmetic progressions then $\lvert A\rvert \leq \exp(-c(\log N)^{1/12})N$ for some constant $c > 0$. Although our proof is identical to that of Kelley and Meka in all of the main ideas, we also incorporate some minor simplifications relating tog Bohr sets. This eases some of the technical difficulties tackled by Kelley and Meka and widens the scope of their method. As a consequence, we improve the lower bounds for finding long arithmetic progressions in $A+A+A$, where $A\subseteq \{1,\ldots,N\}$.

Any rational number can be written as the sum of distinct unit fractions. In this survey paper we review some of the many interesting questions concerning such 'Egyptian fraction' decompositions, and recent progress concerning them.

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 \(\mathbb{F}_q[t]\).