In this talk, I will discuss new bounds on constrained sets of fractions. Specifically, I will discuss the answer to the following question, which arises in several areas of number theory: For an integer $k\geq2$, consider the set of $k$-tuples of reduced fractions $\frac{a1}{q1} , . . . , \frac{ak}{qk} \in I$, where $I$ is an interval around 0. How many $k$-tuples are there with $\sum_{i} \frac{ai}{qi} \in \mathbb{Z} $? When $k$ is even, the answer is well-known: the main contribution to the number of solutions comes from “diagonal” terms, where the fractions $\frac{ai}{qi}$ cancel in pairs. When $k$ is odd, the answer is much more mysterious! In joint work with Bloom, we prove a near-optimal upper bound on this problem when $k$ is odd. I will also discuss applications of this problem to estimating moments of the distributions of primes and reduced residues.[-]

The delta symbol developed by Duke-Friedlander-Iwaniec and Heath-Brown has played an important role in studying rational points on hypersurfaces of low degrees. We present a two dimensional delta symbol and apply it to establish a quantitative Hasse principle for a smooth intersection of two quadratic forms defined over $Q$ in at least ten variables. The goal of these delta symbols is to carry out a (double) Kloosterman refinement of the circle method. This is based on a joint work with Simon Rydin Myerson and Pankaj Vishe.[-]

Let $\lambda$ be the Liouville function, defined by $\lambda(n) = (-1)^{\Omega(n)}$ where $\Omega(n)$ is the number of prime factors of $n$ (with multiplicity). This completely multiplicative Let $\lambda$ be the Liouville function, defined by $\lambda(n)=(-1)^{\Omega(n)}$ where $\Omega(n)$ is the number of prime factors of $n$ (with multiplicity). This completely multiplicative function is believed to exhibit pseudo-random statistical properties. For example, its partial sums are conjectured to obey the square-root cancellation estimate $\sum_{n \leq x} \lambda(n)=O\left(x^{1 / 2+\varepsilon}\right)$; this is equivalent to the Riemann Hypothesis.

The Fourier uniformity conjecture (a close cousin of the Chowla and Sarnak conjectures) concerns the pseudo-random behaviour of the Liouville function in short intervals. In 2023, Walsh proved that, for $\exp \left((\log X)^{1 / 2+\varepsilon}\right) \leq H \leq X$,

$
\sum_{X \lt x \lt 2X} \sup _{\alpha \in \mathbb{R}}\left|\sum_{x\lt n \lt x+H} \lambda(n) e(n \alpha)\right|=o(H X)
$

as $X \rightarrow \infty$. This non-correlation estimate is expected to hold for any $H=H(X)$ tending arbitrarily slowly to infinity with $X$ : this is the Fourier uniformity conjecture.

We improve on Walsh's range, proving that the Fourier uniformity conjecture holds for intervals of length $H \geq \exp \left((\log X)^{2 / 5+\varepsilon}\right)$.[-]

Following Zhang's breakthrough on bounded gaps between primes, much work has gone into improving upper bounds on the smallest integer which appears infinitely often as the gap between a given number of primes. Equidistribution estimates for primes in certain arithmetic progressions are a key ingredient of Zhang's proof and later work of Polymath. In this talk, I will highlight how bounded gaps and primes in arithmetic progressions are linked, and I will discuss obstacles and recent successes in using various types of old and new equistribution estimates to improve on the results of Polymath for bounded gaps between primes.[-]

Given an additive function $f$ and a multiplicative function $g$, let
$E(f,g;x)=\#\left \{ n\leq x:f(n)=g(n) \right \}$
We study the size of $E(f,g;x)$ for those functions $f$ and $g$ such that $f(n)\neq g(n)$ for at least one value of $n> 1$. In particular, when $f(n)=\omega (n)$ , the number of distinct prime factors of $n$ , we show that for any $\varepsilon >0$ , there exists a multiplicative function $g$ such that
$E(\varepsilon ,g;x)\gg \frac{x}{\left ( \log \log x\right )^{1+\varepsilon }}$,
while we prove that $E(\varepsilon ,g;x)=o(x)$ as $x\rightarrow \infty$ for every multiplicative function $g$.[-]

All previous methods of showing the existence of large gaps between primes have relied on the fact that smooth numbers are unusually sparse. This feature of the argument does not seem to generalise to showing large gaps between primes in subsets, such as values of a polynomial. We will talk about recent work which allows us to show large gaps between primes without relying on smooth number estimates. This then generalizes naturally to show long strings of consecutive composite values of a polynomial. This is joint work with Ford, Konyagin, Pomerance and Tao.[-]

Fermat showed that every prime $p = 1$ mod $4$ is a sum of two squares: $p = a^2 + b^2$, and hence such a prime gives rise to an angle whose tangent is the ratio $b/a$. Hecke showed, in 1919, that these angles are uniformly distributed, and uniform distribution in somewhat short arcs was given in by Kubilius in 1950 and refined since then. I will discuss the statistics of these angles on fine scales and present a conjecture, motivated by a random matrix model and by function field considerations.[-]

For a non-principal Dirichlet character $\chi$ modulo $q$, the classical Pólya-Vinogradov inequality asserts that
$M (\chi) := \underset{x}{max}$$| \sum_{n \leq x}$$\chi(n)| = O (\sqrt{q} log$ $q)$.
This was improved to $\sqrt{q} log$ $log$ $q$ by Montgomery and Vaughan, assuming the Generalized Riemann hypothesis GRH. For quadratic characters, this is known to be optimal, owing to an unconditional omega result due to Paley. In this talk, we shall present recent results on higher order character sums. In the first part, we discuss even order characters, in which case we obtain optimal omega results for $M(\chi)$, extending and refining Paley's construction. The second part, joint with Alexander Mangerel, will be devoted to the more interesting case of odd order characters, where we build on previous works of Granville and Soundararajan and of Goldmakher to provide further improvements of the Pólya-Vinogradov and Montgomery-Vaughan bounds in this case. In particular, assuming GRH, we are able to determine the order of magnitude of the maximum of $M(\chi)$, when $\chi$ has odd order $g \geq 3$ and conductor $q$, up to a power of $log_4 q$ (where $log_4$ is the fourth iterated logarithm).[-]

Let $\alpha$ $\epsilon$ $\mathbb{R}^d$ be a vector whose entries $\alpha_1, . . . , \alpha_d$ and $1$ are linearly independent over the rationals. We say that $S \subset \mathbb{T}^d$ is a bounded remainder set for the sequence of irrational rotations $\lbrace n\alpha\rbrace_{n\geqslant1}$ if the discrepancy
$ \sum_{k=1}^{N}1_S (\lbrace k\alpha\rbrace) - N$ $mes(S)$
is bounded in absolute value as $N \to \infty$. In one dimension, Hecke, Ostrowski and Kesten characterized the intervals with this property.
We will discuss the bounded remainder property for sets in higher dimensions. In particular, we will see that parallelotopes spanned by vectors in $\mathbb{Z}\alpha + \mathbb{Z}^d$ have bounded remainder. Moreover, we show that this condition can be established by exploiting a connection between irrational rotation on $\mathbb{T}^d$ and certain cut-and-project sets. If time allows, we will discuss bounded remainder sets for the continuous irrational rotation $\lbrace t \alpha : t$ $\epsilon$ $\mathbb{R}^+\rbrace$ in two dimensions.[-]

We introduce a family of block-additive automatic sequences, that are obtained by allocating a weight to each couple of digits, and defining the nth term of the sequence as being the total weight of the integer n written in base k. Under an additional combinatorial difference condition on the weight function, these sequences can be interpreted as generalised Rudin–Shapiro sequences. We prove that these sequences have the same two-term correlations as sequences of symbols chosen uniformly and independently at random. The speed of convergence is independent of the prime factor decomposition of k. This extends work by E. Grant, J. Shallit, T. Stoll, and by P.-A. Tahay.[-]