If A and B are subsets of an abelian group, their sumset is $A+B:=\{a+b:a\in A, b\in B\}$. We study sumsets in discrete abelian groups, where at least one summand has positive upper Banach density.

Jin proved in [27] that if A and B are sets of integers having positive upper Banach density, then $A+B$ is piecewise syndetic. Bergelson, Furstenberg, and Weiss [4] improved the conclusion to “$A+B$ is piecewise Bohr.” In [2] this was shown to be qualitatively optimal, in the sense that if $C\subseteq \mathbb Z$ is piecewise Bohr, then there are $A, B\subseteq \mathbb Z$ having positive upper Banach density such that $A+B\subseteq C$.

We improve these results by establishing a strong correspondence between sumsets in discrete abelian groups, level sets of convolutions in compact abelian groups, and sumsets in compact abelian groups. Our proofs avoid measure preserving dynamics and nonstandard analysis, and our results apply to discrete abelian groups of any cardinality.

In a previous work, Bettin, Koukoulopoulos, and Sanna prove that if two sets of natural numbers A and B have natural density 1, then their product set $A \cdot B \;:\!=\; \{ab \;:\; a \in A, b \in B\}$ also has natural density 1. They also provide an effective rate and pose the question of determining the optimal rate. We make progress on this question by constructing a set A of density 1 such that $A\cdot A$ has a “large” complement.

Gowers and Hatami initiated the inverse theory for the uniformity norms $U^k$ of matrix-valued functions on non-abelian groups by proving a 1%-inverse theorem for the $U^2$-norm and relating it to stability questions for almost representations. In this paper, we take a step toward an inverse theory for higher-order uniformity norms of matrix-valued functions on arbitrary groups by examining the 99% regime for the $U^k$-norm on perfect groups of bounded commutator width.

This analysis prompts a classification of Leibman’s quadratic maps between non-abelian groups. Our principal contribution is a complete description of these maps via an explicit universal construction. From this classification we deduce several applications: A full classification of quadratic maps on arbitrary abelian groups; a proof that no nontrivial polynomial maps of degree greater than one exist on perfect groups; stability results for approximate polynomial maps.

We prove two results related to the solutions of norm form equations. First, we give a finiteness result for sums of terms of linear recurrence sequences appearing in the coordinates of solutions of norm form equations. Second, we give a finiteness result concerning solutions of norm form equations representable as sums of S-units with a fixed number of terms. To prove these results, we use a deep result concerning the finiteness of the solutions of polynomial-exponential equations and S-unit equations.

We use spectral theory and algebraic geometry to establish a higher-degree analogue of a Szemerédi–Trotter-type theorem over finite fields, with an application to polynomial expansion.

We compute the Galois groups of the reductions modulo a prime number p of the generating series of Apéry numbers, Domb numbers and Almkvist–Zudilin numbers. We observe in particular that their behaviour is governed by congruence conditions on p.

Let $\mathbb {N}$ be the set of all nonnegative integers. For a set $A\subseteq \mathbb {N}$, let $R_2(A,n)$ and $R_3(A,n)$ be the number of solutions of the equation $n=a_1+a_2$ with $a_1<a_2, a_1,a_2\in A$ and with $a_1\le a_2, a_1,a_2\in A$, respectively. If $-N\le g\le N$, Yan [‘On the structure of partition which the difference of their representation function is a constant’, Period. Math. Hungar. 82 (2021), 149–152] showed that there is a set $A\subseteq \mathbb {N}$ such that $R_i(A,n)-R_i(\mathbb {N}\setminus A,n)=g$ for all integers $n\ge 2N-1$, where N is a positive integer. In this paper, we prove that if $g_1,g_2$ are nonnegative integers with $g_1\neq g_2$, then there does not exist $A\subseteq \mathbb {N}$ such that $R_i(A,2n)-R_i(\mathbb {N}\setminus A,2n)=g_1$ and $R_i(A,2n+1)-R_i(\mathbb {N}\setminus A,2n+1)=g_2$ for all sufficiently large integers n.

We show that for any set $A\subset {\mathbb N}$ with positive upper density and any $\ell ,m \in {\mathbb N}$, there exist an infinite set $B\subset {\mathbb N}$ and some $t\in {\mathbb N}$ so that $\{mb_1 + \ell b_2 \colon b_1,b_2\in B\ \text {and}\ b_1<b_2 \}+t \subset A,$ verifying a conjecture of Kra, Moreira, Richter and Robertson. We also consider the patterns $\{mb_1 + \ell b_2 \colon b_1,b_2\in B\ \text {and}\ b_1 \leq b_2 \}$, for infinite $B\subset {\mathbb N}$ and prove that any set $A\subset {\mathbb N}$ with lower density $\underline {\!\mathrm {d}}(A)>1/2$ contains such configurations up to a shift. We show that the value $1/2$ is optimal and obtain analogous results for values of upper density and when no shift is allowed.

We introduce a relaxation of stability, called almost sure stability, which is insensitive to perturbations by subsets of Loeb measure $0$ in a non-standard finite group. We show that almost sure stability satisfies a stationarity principle in the sense of geometric stability theory for measure-independent elements. We apply this principle to deduce the existence of squares in dense almost surely stable subsets of Cartesian products of non-standard finite groups, possibly non-abelian. Our results imply qualitative asymptotic versions for Cartesian products of finite groups. In the final section, we establish the existence of $3\times 2$-grids (and thus of L-shapes) in dense almost surely stable two-dimensional subsets of finite abelian groups of odd order.

Let C and W be two sets of integers. If $C+W=\mathbb {Z}$, then C is called an additive complement to W. We further call C a minimal additive complement to W if no proper subset of C is an additive complement to W. Answering a problem of Nathanson in part, we give sufficient conditions to show that W has no minimal additive complements. Our result extends a result of Chen and Yang [‘On a problem of Nathanson related to minimal additive complements’, SIAM J. Discrete Math. 26 (2012), 1532–1536].

We obtain polylogarithmic bounds in the polynomial Szemerédi theorem when the polynomials have distinct degrees and zero constant terms. Specifically, let $P_1, \dots, P_m \in \mathbb Z[y]$ be polynomials with distinct degrees, each having zero constant term. Then there exists a constant $c = c(P_1,\dots,P_m) \gt 0$ such that any subset $A \subset \{1,2,\dots,N\}$ of density at least $(\log N)^{-c}$ contains a nontrivial polynomial progression of the form $x, x+P_1(y), \dots, x+P_m(y)$. In addition, we prove an effective “popular” version, showing that every dense subset $A$ has some non-zero $y$ such that the number of polynomial progressions in $A$ with this difference $y$ is asymptotically at least as large as in a random set of the same density as $A$.

It is an open problem in additive number theory to compute and understand the full range of sumset sizes of finite sets of integers, that is, the set $ \mathcal R_{\mathbf Z}(h,k) = \{|hA|:A \subseteq \mathbf Z \text { and } |A|=k\}$ for all integers $h \geq 3$ and $k \geq 3$. This article constructs certain infinite families of finite sets of size k, computes their h-fold sumset sizes, and obtains explicit finite arithmetic progressions of sumset sizes in $ \mathcal R_{\mathbf Z}(h,k)$.

We construct large Salem sets avoiding patterns, complementing previous constructions of pattern avoiding sets with large Hausdorff dimension. For a (possibly uncountable) family of uniformly Lipschitz functions $\{f_i:(\mathbb{T}^d)^{n-2} \to \mathbb{T}^d \}$, we obtain a Salem subset of $\mathbb{T}^d$ with dimension $d/(n-1)$ avoiding nontrivial solutions to the equation $x_n-x_{n-1} = f_i(x_1,...,x_{n-2})$. For a countable family of smooth functions $\{f_i: (\mathbb{T}^d)^{n-1} \to \mathbb{T}^d \}$ satisfying a modest geometric condition, we obtain a Salem subset of $\mathbb{T}^d$ with dimension $d/(n-3/4)$ avoiding nontrivial solutions to the equation $x_n= f(x_1,...,x_{n-1})$. For a set $Z \subset \mathbb{T}^{dn}$ which is the countable union of a family of sets, each with lower Minkowski dimension $s$, we obtain a Salem subset of $\mathbb{T}^d$ of dimension $(dn-s)/(n-1/2)$ whose Cartesian product does not intersect $Z$ except at points with non-distinct coordinates.

Let A be a set of natural numbers. A set B of natural numbers is an additive complement of the set A if all sufficiently large natural numbers can be represented in the form $x+y$, where $x\in A$ and $y\in B$. We establish that if $A=\{a_i: i\in \mathbb {N}\}$ is a set of natural numbers such that $a_i<a_{i+1} $ for $i \in \mathbb {N}$ and $\liminf _{n\rightarrow \infty } (a_{n+1}/a_{n})>1$, then there exists a set $B\subset \mathbb {N}$ such that $B\cap A = \varnothing $ and B is a sparse additive complement of the set A.

A system of linear equations in $\mathbb {F}_p^n$ is Sidorenko if any subset of $\mathbb {F}_p^n$ contains at least as many solutions to the system as a random set of the same density, asymptotically as $n\to \infty $. A system of linear equations is common if any two-colouring of $\mathbb {F}_p^n$ yields at least as many monochromatic solutions to the system of equations as a random 2-colouring, asymptotically as $n\to \infty $. Both classification problems remain wide open despite recent attention.

We show that a certain generic family of systems of two linear equations is not Sidorenko. In fact, we show that systems in this family are not locally Sidorenko, and that systems in this family which do not contain additive tuples are not weakly locally Sidorenko. This endeavour answers a conjecture and question of Kamčev–Liebenau–Morrison. Insofar as methods, we observe that the true complexity of a linear system is not maintained under Fourier inversion; our main novelty is the use of higher-order methods in the frequency space of systems which have complexity one. We also give a shorter proof of the recent result of Kamčev–Liebenau–Morrison and independently Versteegen that any linear system containing a four-term arithmetic progression is uncommon.

Let $\{u_n\}_n$ be a nondegenerate linear recurrence sequence of integers with Binet’s formula given by ${u_n= \sum _{i=1}^{m} P_i(n)\alpha _i^n.}$ Assume $\max _i \vert \alpha _i \vert>1$. In 1977, Loxton and Van der Poorten conjectured that for any $\epsilon>0$, there is an effectively computable constant $C(\epsilon )$ such that if $ \vert u_n \vert < (\max _i\{ \vert \alpha _i \vert \})^{n(1-\epsilon )}$, then $n<C(\epsilon )$. Using results of Schmidt and Evertse, a complete noneffective (qualitative) proof of this conjecture was given by Fuchs and Heintze [‘On the growth of linear recurrences in function fields’, Bull. Aust. Math. Soc. 104(1) (2021), 11–20] and, independently, by Karimov et al. [‘The power of positivity’, Proc. LICS 23 (2023), 1–11]. In this paper, we give an effective upper bound for the number of solutions of the inequality $\vert u_n \vert < (\max _i\{ \vert \alpha _i \vert \})^{n(1-\epsilon )}$, thus extending several earlier results by Schmidt, Schlickewei and Van der Poorten.

We consider a new family $(\mathcal {T}_n)_{n\geq 1}$ of aperiodic sets of Wang tiles and we describe the dynamical properties of the set $\Omega _n$ of valid configurations $\mathbb {Z}^2\to \mathcal {T}_n$. The tiles can be defined as the different instances of a square-shaped computer chip whose inputs and outputs are 3-dimensional integer vectors. The family include the Ammann aperiodic set of 16 Wang tiles and gathers the hallmarks of other small aperiodic sets of Wang tiles. Notably, the tiles satisfy additive versions of equations verified by the Kari–Culik aperiodic sets of 14 and 13 Wang tiles. Also configurations in $\Omega _n$ are the codings of a $\mathbb {Z}^2$-action on a 2-dimensional torus like the Jeandel–Rao aperiodic set of 11 Wang tiles. The family broadens the relation between quadratic integers and aperiodic tilings beyond the omnipresent golden ratio as the dynamics of $\Omega _n$ involves the positive root $\beta $ of the polynomial $x^2-nx-1$, also known as the n-th metallic mean. We show the existence of an almost one-to-one factor map $\Omega _n\to \mathbb {T}^2$ which commutes the shift action on $\Omega _n$ with horizontal and vertical translations by $\beta $ on $\mathbb {T}^2$. The factor map can be explicitly defined by the average of the top labels from the same row of tiles as in Kari and Culik examples. The proofs are based on the minimality of $\Omega _n$ (proved in a previous article) and a polygonal partition of $\mathbb {T}^2$ which we show is a Markov partition for the toral $\mathbb {Z}^2$-action. The partition and the sets of Wang tiles are symmetric which makes them, like Penrose tilings, worthy of investigation.

We present an explicit subset $A\subseteq \mathbb{N} = \{0,1,\ldots \}$ such that $A + A = \mathbb{N}$ and for all $\varepsilon \gt 0$,

\begin{equation*}\lim _{N\to \infty }\frac {\big |\big \{(n_1,n_2): n_1 + n_2 = N, (n_1,n_2)\in A^2\big \}\big |}{N^{\varepsilon }} = 0.\end{equation*}

This answers a question of Erdős.

The paperfolding sequences form an uncountable class of infinite sequences over the alphabet $\{ -1, 1 \}$ that describe the sequence of folds arising from iterated folding of a piece of paper, followed by unfolding. In this note, we observe that the sequence of run lengths in such a sequence, as well as the starting and ending positions of the nth run, is $2$-synchronised and hence computable by a finite automaton. As a specific consequence, we obtain the recent results of Bunder, Bates and Arnold [‘The summed paperfolding sequence’, Bull. Aust. Math. Soc. 110 (2024), 189–198] in much more generality, via a different approach. We also prove results about the critical exponent and subword complexity of these run-length sequences.

We show that on a $\sigma $-finite measure-preserving system $X = (X,\nu , T)$, the non-conventional ergodic averages

$$ \begin{align*} \mathbb{E}_{n \in [N]} \Lambda(n) f(T^n x) g(T^{P(n)} x) \end{align*} $$

$f \in L^{p_1}(X)$

$g \in L^{p_2}(X)$

$1/p_1 + 1/p_2 \leq 1$

$2$

$\Lambda $

$1$

$\mu $