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 $

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.

Kam Cheong Au [‘Wilf–Zeilberger seeds and non-trivial hypergeometric series’, Journal of Symbolic Computation 130 (2025), Article no. 102241] discovered a powerful methodology for finding new Wilf–Zeilberger (WZ) pairs. He calls it WZ seeds and gives numerous examples of applications to proving longstanding conjectural identities for reciprocal powers of $\pi $ and their duals for Dirichlet L-values. In this note, we explain how a modification of Au’s WZ pairs together with a classical analytic argument yields simpler proofs of these results. We illustrate our method with examples elaborated with assistance of Maple code that we have developed.

For a wide class of integer linear recurrence sequences $(u(n))_{n=1}^\infty $, we give an upper bound on the number of s-tuples $\left (n_1, \ldots , n_s\right ) \in \left ({\mathbb Z}\cap [M+1,M+ N]\right )^s$ such that the corresponding elements $u(n_1), \ldots , u(n_s)$ in the sequence are multiplicatively dependent.

We study linear random walks on the torus and show a quantitative equidistribution statement, under the assumption that the Zariski closure of the acting group is semisimple.

We construct skew corner-free subsets of $[n]^2$ of size $n^2\exp(\!-O(\sqrt{\log n}))$, thereby improving on recent bounds of the form $\Omega(n^{5/4})$ obtained by Pohoata and Zakharov. We also prove that any such set has size at most $O(n^2(\log n)^{-c})$ for some absolute constant $c \gt 0$. This improves on the previously best known upper bound $O(n^2(\log\log n)^{-c})$, coming from Shkredov’s work on the corners theorem.

It was asked by E. Szemerédi if, for a finite set $A\subset {\mathbb {Z}}$, one can improve estimates for $\max \{|A+A|,|A\cdot A|\}$, under the constraint that all integers involved have a bounded number of prime factors, that is, each $a\in A$ satisfies $\omega (a)\leq k$. In this paper we show that this maximum is at least of order $|A|^{\frac {5}{3}-o_\epsilon (1)}$ provided $k\leq (\log |A|)^{1-\varepsilon }$ for any $\varepsilon \gt 0$. In fact, this will follow from an estimate for additive energy which is best possible up to factors of size $|A|^{o(1)}$.

Green and Tao’s arithmetic regularity lemma and counting lemma together apply to systems of linear forms which satisfy a particular algebraic criterion known as the ‘flag condition’. We give an arithmetic regularity lemma and counting lemma which apply to all systems of linear forms.

In this article, we investigate the multiplicative structure of a shifted multiplicative subgroup and its connections with additive combinatorics and the theory of Diophantine equations. Among many new results, we highlight our main contributions as follows. First, we show that if a nontrivial shift of a multiplicative subgroup G contains a product set $AB$, then $|A||B|$ is essentially bounded by $|G|$, refining a well-known consequence of a classical result by Vinogradov. Second, we provide a sharper upper bound of $M_k(n)$, the largest size of a set such that each pairwise product of its elements is n less than a kth power, refining the recent result of Dixit, Kim, and Murty. One main ingredient in our proof is the first non-trivial upper bound on the maximum size of a generalized Diophantine tuple over a finite field. In addition, we determine the maximum size of an infinite family of generalized Diophantine tuples over finite fields with square order, which is of independent interest. We also make significant progress toward a conjecture of Sárközy on the multiplicative decompositions of shifted multiplicative subgroups. In particular, we prove that for almost all primes p, the set $\{x^2-1: x \in {\mathbb F}_p^*\} \setminus \{0\}$ cannot be decomposed as the product of two sets in ${\mathbb F}_p$ non-trivially.