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Let $A \subset \mathbb{Z}^d$ be a finite set. It is known that the sumset $NA$ has predictable size ($\vert NA\vert = P_A(N)$ for some $P_A(X) \in \mathbb{Q}[X]$) and structure (all of the lattice points in some finite cone other than all of the lattice points in a finite collection of exceptional subcones), once $N$ is larger than some threshold. In previous work, joint with Shakan, the first and third named authors established the first effective bounds for both of these thresholds for an arbitrary set $A$. In this article we substantially improve each of these bounds, coming much closer to the corresponding lower bounds known., Comment: 15 pages

We present a novel proof of the Duffin-Schaeffer conjecture in metric Diophantine approximation. Our proof is heavily motivated by the ideas of Koukoulopoulos-Maynard's breakthrough first argument, but simplifies and strengthens several technical aspects. In particular, we avoid any direct handling of GCD graphs and their `quality'. We also consider the metric quantitative theory of Diophantine approximations, improving the $(\log \Psi(N))^{-C}$ error-term of Aistleitner-Borda and the first named author to $\exp(-(\log \Psi(N))^{\frac{1}{2} - \varepsilon})$.

Let $\lambda$ denote the Liouville function. We show that the logarithmic mean of $\lambda(\lfloor \alpha_1n\rfloor)\lambda(\lfloor \alpha_2n\rfloor)$ is $0$ whenever $\alpha_1,\alpha_2$ are positive reals with $\alpha_1/\alpha_2$ irrational. We also show that for $k\geq 3$ the logarithmic mean of $\lambda(\lfloor \alpha_1n\rfloor)\cdots \lambda(\lfloor \alpha_kn\rfloor)$ has some nontrivial amount of cancellation, under certain rational independence assumptions on the real numbers $\alpha_i$. Our results for the Liouville function generalise to produce independence statements for general bounded real-valued multiplicative functions evaluated at Beatty sequences. These results answer the two-point case of a conjecture of Frantzikinakis (and provide some progress on the higher order cases), generalising a recent result of Crn\v{c}evi\'c--Hern\'andez--Rizk--Sereesuchart--Tao. As an ingredient in our proofs, we establish bounds for the logarithmic correlations of the Liouville function along Bohr sets.

Let $A \subset \mathbb{Z}^d$ be a finite set. It is known that $NA$ has a particular size ($\vert NA\vert = P_A(N)$ for some $P_A(X) \in \mathbb{Q}[X]$) and structure (all of the lattice points in a cone other than certain exceptional sets), once $N$ is larger than some threshold. In this article we give the first effective upper bounds for this threshold for arbitrary $A$. Such explicit results were only previously known in the special cases when $d=1$, when the convex hull of $A$ is a simplex or when $\vert A\vert = d+2$, results which we improve., Comment: Small improvements to simplex case, following referee suggestions. To appear in Combinatorica

In this note we give two small results concerning the correlations of the Selberg sieve weights. We then use these estimates to derive a new (conditional) lower bound on the variance of the primes in short intervals, and also on the so-called `form factor' for the pair correlations of the zeros of the Riemann zeta function. Our bounds ultimately rely on the estimates of Bettin--Chandee for trilinear Kloosterman fractions., Comment: Correction to some exponents

We prove that if $A \subseteq [X, 2X]$ and $B \subseteq [Y, 2Y]$ are sets of integers such that $\gcd(a,b) \geq D$ for at least $\delta |A||B|$ pairs $(a,b) \in A \times B$ then $|A||B| \ll_{\varepsilon} \delta^{-2 - \varepsilon} XY/D^2$. This is a new result even when $\delta = 1$. The proof uses ideas of Koukoulopoulos and Maynard and some additional combinatorial arguments., Comment: One remark added

Let $A = \{0 = a_0 < a_1 < \cdots < a_{\ell + 1} = b\}$ be a finite set of non-negative integers. We prove that the sumset $NA$ has a certain easily-described structure, provided that $N \geqslant b-\ell$, as recently conjectured by Shakan and the first author. We also classify those sets $A$ for which this bound cannot be improved., Comment: 8 pages, minor changes to exposition. Accepted version

For fixed $\alpha \in [0,1]$, consider the set $S_{\alpha,N}$ of dilated squares $\alpha, 4\alpha, 9\alpha, \dots, N^2\alpha \, $ modulo $1$. Rudnick and Sarnak conjectured that for Lebesgue almost all such $\alpha$ the gap-distribution of $S_{\alpha,N}$ is consistent with the Poisson model (in the limit as $N$ tends to infinity). In this paper we prove a new estimate for the triple correlations associated to this problem, establishing an asymptotic expression for the third moment of the number of elements of $S_{\alpha,N}$ in a random interval of length $L/N$, provided that $L > N^{1/4+\varepsilon}$. The threshold of $1/4$ is substantially smaller than the threshold of $1/2$ (which is the threshold that would be given by a na\"{i}ve discrepancy estimate). Unlike the theory of pair correlations, rather little is known about triple correlations of the dilations $(\alpha a_n \, \text{mod } 1)_{n=1}^{\infty} $ for a non-lacunary sequence $(a_n)_{n=1}^{\infty} $ of increasing integers. This is partially due to the fact that second moment of the triple correlation function is difficult to control, and thus standard techniques involving variance bounds are not applicable. We circumvent this impasse by using an argument inspired by works of Rudnick--Sarnak--Zaharescu and Heath-Brown, which connects the triple correlation function to some modular counting problems. In an appendix we comment on the relationship between discrepancy and correlation functions, answering a question of Steinerberger., Comment: Final version. To appear in Canadian Journal of Mathematics

In this paper we prove an asymptotic formula for the number of solutions in prime numbers to systems of simultaneous linear inequalities with algebraic coefficients. For $m$ simultaneous inequalities we require at least $m+2$ variables, improving upon existing methods, which generically require at least $2m+1$ variables. Our result also generalises the theorem of Green-Tao-Ziegler on linear equations in primes. Many of the methods presented apply for arbitrary coefficients, not just for algebraic coefficients, and we formulate a conjecture concerning the pseudorandomness of sieve weights which, if resolved, would remove the algebraicity assumption entirely., Comment: 71 pages. Minor corrections from version 1. Accepted for publication in Journal d'Analyse Math\'ematique

For an integer $b \geqslant 2$ and a set $S\subset \{0,\cdots,b-1\}$, we define the Kempner set $\mathcal{K}(S,b)$ to be the set of all non-negative integers whose base-$b$ digital expansions contain only digits from $S$. These well-studied sparse sets provide a rich setting for additive number theory, and in this paper we study various questions relating to the appearance of arithmetic progressions in these sets. In particular, for all $b$ we determine exactly the maximal length of an arithmetic progression that omits a base-$b$ digit., Comment: 11 pages, submitted to American Mathematical Monthly

In this note we prove a new estimate on so-called GCD sums (also called G\'{a}l sums), which, for certain coefficients, improves significantly over the general bound due to de la Bret\`{e}che 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., Comment: 7 pages

Let $A$ be a set of natural numbers. Recent work has suggested a strong link between the additive energy of $A$ (the number of solutions to $a_1 + a_2 = a_3 + a_4$ with $a_i \in A$) and the metric Poissonian property, which is a fine-scale equidistribution property for dilates of $A$ modulo $1$. There appears to be reasonable evidence to speculate a sharp Khintchine-type threshold, that is, to speculate that the metric Poissonian property should be completely determined by whether or not a certain sum of additive energies is convergent or divergent. In this article, we primarily address the convergence theory, in other words the extent to which having a low additive energy forces a set to be metric Poissonian., Comment: Slight changes from version 1 based on comments from the referee

A central tool in the study of systems of linear equations with integer coefficients is the Generalised von Neumann Theorem of Green and Tao. This theorem reduces the task of counting the weighted solutions of these equations to that of counting the weighted solutions for a particular family of forms, the Gowers norms $\Vert f \Vert_{U^{s+1}[N]}$ of the weight $f$. In this paper we consider systems of linear inequalities with real coefficients, and show that the number of solutions to such weighted diophantine inequalities may also be bounded by Gowers norms. Furthermore, we provide a necessary and sufficient condition for a system of real linear forms to be governed by Gowers norms in this way. We present applications to cancellation of the M\"{o}bius function over certain sequences. The machinery developed in this paper can be adapted to the case in which the weights are unbounded but suitably pseudorandom, with applications to counting the number of solutions to diophantine inequalities over the primes. Substantial extra difficulties occur in this setting, however, and we have prepared a separate paper on these issues., Comment: 75 pages. Reworked introduction based on referee comments. To appear in Algebra & Number Theory

It has been known since Vinogradov that, for irrational $\alpha$, the sequence of fractional parts $\{\alpha p\}$ is equidistributed in $\mathbb{R}/\mathbb{Z}$ as $p$ ranges over primes. There is a natural second-order equidistribution property, a pair correlation of such fractional parts, which has recently received renewed interest, in particular regarding its relation to additive combinatorics. In this paper we show that the primes do not enjoy this stronger equidistribution property., Comment: 5 pages + references. Small corrections from version 1

We prove that, if $x$ and $q\leqslant x^{1/16}$ are two parameters, then for any invertible residue class $a$ modulo $q$ there exists a product of exactly three primes, each one below $x^{1/3}$, that is congruent to $a$ modulo $q$., Comment: 6 pages, to submitted to Journal de Th\'{e}orie des Nombres de Bordeaux. Small corrections to previous version

The classical theorem of Schnirelmann states that the primes are an additive basis for the integers. In this paper we consider the analogous multiplicative setting of the cyclic group $\left(\mathbb{Z}/ q\mathbb{Z}\right)^{\times}$, and prove a similar result. For all suitably large primes $q$ we define $P_\eta$ to be the set of primes less than $\eta q$, viewed naturally as a subset of $\left(\mathbb{Z}/ q\mathbb{Z}\right)^{\times}$. Considering the $k$-fold product set $P_\eta^{(k)}=\{p_1p_2\cdots p_k:p_i\in P_\eta \}$, we show that for $\eta \gg q^{-\frac{1}{4}+\epsilon}$ there exists a constant $k$ depending only on $\epsilon$ such that $P_\eta^{(k)}=\left(\mathbb{Z}/ q\mathbb{Z}\right)^{\times}$. Erd\H{o}s conjectured that for $\eta = 1$ the value $k=2$ should suffice: although we have not been able to prove this conjecture, we do establish that $P_1 ^{(2)}$ has density at least $\frac{1}{64}(1+o(1))$. We also formulate a similar theorem in almost-primes, improving on existing results., Comment: 10 pages + references. Small corrections in light of comments from referee. To appear in Bulletin of the London Mathematical Society