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.

We address a core partition regularity problem in Ramsey theory by proving that every finite coloring of the positive integers contains monochromatic Pythagorean pairs (i.e., $x,y\in {\mathbb N}$ such that $x^2\pm y^2=z^2$ for some $z\in {\mathbb N}$). We also show that partitions generated by level sets of multiplicative functions taking finitely many values always contain Pythagorean triples. Our proofs combine known Gowers uniformity properties of aperiodic multiplicative functions with a novel and rather flexible approach based on concentration estimates of multiplicative functions.

Let C and W be two integer sets. If $C+W=\mathbb {Z}$, then we say that C is an additive complement to W. If no proper subset of C is an additive complement to W, then we say that C is a minimal additive complement to W. We study the existence of a minimal additive complement to $W=\{w_i\}_{i=1}^{\infty}$ when W is not eventually periodic and $w_{i+1}-w_{i}\in \{2,3\}$ for all i.

The triangle removal states that if G contains $\varepsilon n^2$ edge-disjoint triangles, then G contains $\delta (\varepsilon )n^3$ triangles. Unfortunately, there are no sensible bounds on the order of growth of $\delta (\varepsilon )$, and at any rate, it is known that $\delta (\varepsilon )$ is not polynomial in $\varepsilon $. Csaba recently obtained an asymmetric variant of the triangle removal, stating that if G contains $\varepsilon n^2$ edge-disjoint triangles, then G contains $2^{-\operatorname {\mathrm {poly}}(1/\varepsilon )}\cdot n^5$ copies of $C_5$. To this end, he devised a new variant of Szemerédi’s regularity lemma. We obtain the following results:

• • We first give a regularity-free proof of Csaba’s theorem, which improves the number of copies of $C_5$ to the optimal number $\operatorname {\mathrm {poly}}(\varepsilon )\cdot n^5$.

• • We say that H is $K_3$-abundant if every graph containing $\varepsilon n^2$ edge-disjoint triangles has $\operatorname {\mathrm {poly}}(\varepsilon )\cdot n^{\lvert V(H)\rvert }$ copies of H. It is easy to see that a $K_3$-abundant graph must be triangle-free and tripartite. Given our first result, it is natural to ask if all triangle-free tripartite graphs are $K_3$-abundant. Our second result is that assuming a well-known conjecture of Ruzsa in additive number theory, the answer to this question is negative.

Our proofs use a mix of combinatorial, number-theoretic, probabilistic and Ramsey-type arguments.

Let $\mathbb{N}$ be the set of all non-negative integers. For any integer r and m, let $r+m\mathbb{N}=\{r+mk: k\in\mathbb{N}\}$. For $S\subseteq \mathbb{N}$ and $n\in \mathbb{N}$, let $R_{S}(n)$ denote the number of solutions of the equation $n=s+s'$ with $s, s'\in S$ and $s \lt s'$. Let $r_{1}, r_{2}, m$ be integers with $0 \lt r_{1} \lt r_{2} \lt m$ and $2\mid r_{1}$. In this paper, we prove that there exist two sets C and D with $C\cup D=\mathbb{N}$ and $C\cap D=(r_{1}+m\mathbb{N})\cup (r_{2}+m\mathbb{N})$ such that $R_{C}(n)=R_{D}(n)$ for all $n\in\mathbb{N}$ if and only if there exists a positive integer l such that $r_{1}=2^{2l+1}-2, r_{2}=2^{2l+1}-1, m=2^{2l+2}-2$.