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 $

We prove that the structure $(\mathbb {Z},<,+,R)$ is distal for all congruence-periodic sparse predicates $R\subseteq \mathbb {N}$. We do so by constructing a strong honest definition for every formula $\phi (x;y)$ with $\lvert {x}\rvert =1$, providing a rare example of concrete distal decompositions.

For every positive integer n, we introduce a set ${\mathcal {T}}_n$ made of $(n+3)^2$ Wang tiles (unit squares with labeled edges). We represent a tiling by translates of these tiles as a configuration $\mathbb {Z}^2\to {\mathcal {T}}_n$. A configuration is valid if the common edge of adjacent tiles has the same label. For every $n\geq 1$, we show that the Wang shift ${\Omega }_n$, defined as the set of valid configurations over the tiles ${\mathcal {T}}_n$, is self-similar, aperiodic and minimal for the shift action. We say that $\{{\Omega }_n\}_{n\geq 1}$ is a family of metallic mean Wang shifts, since the inflation factor of the self-similarity of $\Omega _n$ is the positive root of the polynomial $x^2-nx-1$. This root is sometimes called the n-th metallic mean, and in particular, the golden mean when $n=1$, and the silver mean when $n=2$. When $n=1$, the set of Wang tiles ${\mathcal {T}}_1$ is equivalent to the Ammann aperiodic set of 16 Wang tiles.

We show that the set of Liouville numbers has a rich set-theoretic structure: it can be partitioned in an explicit way into an uncountable collection of subsets, each of which is dense in the real line. Furthermore, each of these partitioning subsets can be similarly partitioned, and the process can be repeated indefinitely.

Given any finite collection $\mathcal {G}$ of translation-invariant linear equations of the form

$$ \begin{align*} \sum_{i=1}^{v} m_i x_i = m_0 x_0, \end{align*} $$

where $(m_0, m_1, \ldots , m_v) \in \mathbb {N}^{v+1}$, $m_0 = \sum _{i=1}^{v} m_i$ and $v \ge 2$, we estimate lower and upper bounds of the supremum of the Hausdorff dimension of sets on the real line that uniformly avoid nontrivial zeros of any f in $\mathcal {G}$.

We provide numerical evidence towards three conjectures on harmonic numbers by Eswarathasan, Levine and Boyd. Let $J_p$ denote the set of integers $n\geq 1$ such that the harmonic number $H_n$ is divisible by a prime p. The conjectures state that: (i) $J_p$ is always finite and of the order $O(p^2(\log \log p)^{2+\epsilon })$; (ii) the set of primes for which $J_p$ is minimal (called harmonic primes) has density $e^{-1}$ among all primes; (iii) no harmonic number is divisible by $p^4$. We prove parts (i) and (iii) for all $p\leq 16843$ with at most one exception, and enumerate harmonic primes up to $50\times 10^5$, finding a proportion close to the expected density. Our work extends previous computations by Boyd by a factor of approximately $30$ and $50$, respectively.

We study optimal dimensionless inequalities

\begin{equation*} \|f\|_{\textrm{U}^k} \leqslant \|f\|_{\ell^{p_{k,n}}} \end{equation*}

that hold for all functions $f\colon\mathbb{Z}^d\to\mathbb{C}$ supported in $\{0,1,\ldots,n-1\}^d$ and estimates

\begin{equation*} \|\mathbb{1}_A\|_{\textrm{U}^k}^{2^k}\leqslant |A|^{t_{k,n}} \end{equation*}

that hold for all subsets A of the same discrete cubes. A general theory, analogous to the work of de Dios Pont, Greenfeld, Ivanisvili, and Madrid, is developed to show that the critical exponents are related by $p_{k,n} t_{k,n} = 2^k$. This is used to prove the three main results of the article:

• • an explicit formula for $t_{k,2}$, which generalizes a theorem by Kane and Tao,

• • two-sided asymptotic estimates for $t_{k,n}$ as $n\to\infty$ for a fixed $k\geqslant2$, which generalize a theorem by Shao, and

• • a precise asymptotic formula for $t_{k,n}$ as $k\to\infty$ for a fixed $n\geqslant2$.

Let $K = \mathbf {R}$ or $\mathbf {C}$. An n-element subset A of K is a $B_h$-set if every element of K has at most one representation as the sum of h not necessarily distinct elements of A. Associated with the $B_h$-set $A = \{a_1,\ldots , a_n\}$ are the $B_h$-vectors $\mathbf {a} = (a_1,\ldots , a_n)$ in $K^n$. This article proves that “almost all” n-element subsets of K are $B_h$-sets in the sense that the set of all $B_h$-vectors is a dense open subset of $K^n$.

A class of sequences called L-sequences is introduced, each one being a subsequence of a Collatz sequence. Every ordered pair $(v,w)$ of positive integers determines an odd positive integer P such that there exists an L-sequence of length n for every positive integer n, each term of which is congruent to P modulo $2^{v+w+1}$. The smallest possible initial term of such a sequence is described. If $3^v>2^{v+w}$ the L-sequence is increasing. Otherwise, it is decreasing, except if it is the constant sequence P. A central role is played by Bezout’s identity.

Ruzsa asked whether there exist Fourier-uniform subsets of $\mathbb Z/N\mathbb Z$ with density $\alpha$ and 4-term arithmetic progression (4-AP) density at most $\alpha^C$, for arbitrarily large C. Gowers constructed Fourier uniform sets with density $\alpha$ and 4-AP density at most $\alpha^{4+c}$ for some small constant $c \gt 0$. We show that an affirmative answer to Ruzsa’s question would follow from the existence of an $N^{o(1)}$-colouring of [N] without symmetrically coloured 4-APs. For a broad and natural class of constructions of Fourier-uniform subsets of $\mathbb Z/N\mathbb Z$, we show that Ruzsa’s question is equivalent to our arithmetic Ramsey question.

We prove analogous results for all even-length APs. For each odd $k\geq 5$, we show that there exist $U^{k-2}$-uniform subsets of $\mathbb Z/N\mathbb Z$ with density $\alpha$ and k-AP density at most $\alpha^{c_k \log(1/\alpha)}$. We also prove generalisations to arbitrary one-dimensional patterns.