We derive a q-supercongruence modulo the third power of a cyclotomic polynomial with the help of Guo and Zudilin’s method of creative microscoping [‘A q-microscope for supercongruences’, Adv. Math. 346 (2019), 329–358] and the q-Dixon formula. As consequences, we give several supercongruences including

$$ \begin{align*}\sum_{k=0}^{(p-2)/3}\frac{(\frac{2}{3})_k^3}{(1)_k^3}\equiv\frac{p}{2}\frac{(1)_{(p-2)/3}}{(\frac{4}{3})_{(p-2)/3}}\pmod{p^3},\end{align*} $$

where p is a prime with $p\equiv 5\pmod {6}$.

We show that there is a red-blue colouring of $[N]$ with no blue 3-term arithmetic progression and no red arithmetic progression of length $e^{C(\log N)^{3/4}(\log \log N)^{1/4}}$. Consequently, the two-colour van der Waerden number $w(3,k)$ is bounded below by $k^{b(k)}$, where $b(k) = c \big ( \frac {\log k}{\log \log k} \big )^{1/3}$. Previously it had been speculated, supported by data, that $w(3,k) = O(k^2)$.

Recently, Lin and Liu [‘Congruences for the truncated Appell series $F_3$ and $F_4$’, Integral Transforms Spec. Funct. 31(1) (2020), 10–17] confirmed a supercongruence on the truncated Appell series $F_3$. Motivated by their work, we give a generalisation of this supercongruence by establishing a q-supercongruence modulo the fourth power of a cyclotomic polynomial.

When a page, represented by the interval $[0,1]$, is folded right over left $n $ times, the right-hand fold contains a sequence of points. We specify these points using two different representation techniques, both involving binary signed-digit representations.

We establish the mean convergence for multiple ergodic averages with iterates given by distinct fractional powers of primes and related multiple recurrence results. A consequence of our main result is that every set of integers with positive upper density contains patterns of the form $\{m,m+[p_n^a], m+[p_n^b]\}$, where $a,b$ are positive nonintegers and $p_n$ denotes the nth prime, a property that fails if a or b is a natural number. Our approach is based on a recent criterion for joint ergodicity of collections of sequences, and the bulk of the proof is devoted to obtaining good seminorm estimates for the related multiple ergodic averages. The input needed from number theory are upper bounds for the number of prime k-tuples that follow from elementary sieve theory estimates and equidistribution results of fractional powers of primes in the circle.

Let n and k be positive integers with $n\ge k+1$ and let $\{a_i\}_{i=1}^n$ be a strictly increasing sequence of positive integers. Let $S_{n, k}:=\sum _{i=1}^{n-k} {1}/{\mathrm {lcm}(a_{i},a_{i+k})}$. In 1978, Borwein [‘A sum of reciprocals of least common multiples’, Canad. Math. Bull. 20 (1978), 117–118] confirmed a conjecture of Erdős by showing that $S_{n,1}\le 1-{1}/{2^{n-1}}$. Hong [‘A sharp upper bound for the sum of reciprocals of least common multiples’, Acta Math. Hungar. 160 (2020), 360–375] improved Borwein’s upper bound to $S_{n,1}\le {a_{1}}^{-1}(1-{1}/{2^{n-1}})$ and derived optimal upper bounds for $S_{n,2}$ and $S_{n,3}$. In this paper, we present a sharp upper bound for $S_{n,4}$ and characterise the sequences $\{a_i\}_{i=1}^n$ for which the upper bound is attained.

We give a new q-analogue of the (A.2) supercongruence of Van Hamme. Our proof employs Andrews’ multiseries generalisation of Watson’s $_{8}\phi _{7}$ transformation, Andrews’ terminating q-analogue of Watson’s $_{3}F_{2}$ summation, a q-Watson-type summation due to Wei–Gong–Li and the creative microscoping method, developed by the author and Zudilin [‘A q-microscope for supercongruences’, Adv. Math. 346 (2019), 329–358]. As a conclusion, we confirm a weaker form of Conjecture 4.5 by the author [‘Some generalizations of a supercongruence of van Hamme’, Integral Transforms Spec. Funct. 28 (2017), 888–899].

Swisher [‘On the supercongruence conjectures of van Hamme’, Res. Math. Sci. 2 (2015), Article no. 18] and He [‘Supercongruences on truncated hypergeometric series’, Results Math. 72 (2017), 303–317] independently proved that Van Hamme’s (G.2) supercongruence holds modulo $p^4$ for any prime $p\equiv 1\pmod {4}$. Swisher also obtained an extension of Van Hamme’s (G.2) supercongruence for $p\equiv 3 \pmod 4$ and $p>3$. In this note, we give new one-parameter generalisations of Van Hamme’s (G.2) supercongruence modulo $p^3$ for any odd prime p. Our proof uses the method of ‘creative microscoping’ introduced by Guo and Zudilin [‘A q-microscope for supercongruences’, Adv. Math. 346 (2019), 329–358].

Recently Ovsienko and Tabachnikov considered extensions of Somos and Gale-Robinson sequences, defined over the algebra of dual numbers. Ovsienko used the same idea to construct so-called shadow sequences derived from other nonlinear recurrence relations exhibiting the Laurent phenomenon, with the original motivation being the hope that these examples should lead to an appropriate notion of a cluster superalgebra, incorporating Grassmann variables. Here, we present various explicit expressions for the shadow of Somos-4 sequences and describe the solution of a general Somos-4 recurrence defined over the $\mathbb{C}$-algebra of dual numbers from several different viewpoints: analytic formulae in terms of elliptic functions, linear difference equations, and Hankel determinants.

By combining the generating function approach with the Lagrange expansion formula, we evaluate, in closed form, two multiple alternating sums of binomial coefficients, which can be regarded as alternating counterparts of the circular sum evaluation discovered by Carlitz [‘The characteristic polynomial of a certain matrix of binomial coefficients’, Fibonacci Quart. 3(2) (1965), 81–89].

We show that the $Kn$–smooth part of $a^n-1$ for an integer $a>1$ is $a^{o(n)}$ for most positive integers n.

A finite set of integers A tiles the integers by translations if $\mathbb {Z}$ can be covered by pairwise disjoint translated copies of A. Restricting attention to one tiling period, we have $A\oplus B=\mathbb {Z}_M$ for some $M\in \mathbb {N}$ and $B\subset \mathbb {Z}$. This can also be stated in terms of cyclotomic divisibility of the mask polynomials $A(X)$ and $B(X)$ associated with A and B.

In this article, we introduce a new approach to a systematic study of such tilings. Our main new tools are the box product, multiscale cuboids and saturating sets, developed through a combination of harmonic-analytic and combinatorial methods. We provide new criteria for tiling and cyclotomic divisibility in terms of these concepts. As an application, we can determine whether a set A containing certain configurations can tile a cyclic group $\mathbb {Z}_M$, or recover a tiling set based on partial information about it. We also develop tiling reductions where a given tiling can be replaced by one or more tilings with a simpler structure. The tools introduced here are crucial in our proof in [24] that all tilings of period $(pqr)^2$, where $p,q,r$ are distinct odd primes, satisfy a tiling condition proposed by Coven and Meyerowitz [2].

We consider sums involving the divisor function over nonhomogeneous ($\beta \neq 0$) Beatty sequences $ \mathcal {B}_{\alpha ,\beta }:=\{[\alpha n+\beta ]\}_{n=1}^{\infty } $ and show that

$$ \begin{align*} \sum_{n\leq N,\ n\in\mathcal{B}_{\alpha,\beta}}d(n) =\alpha^{-1}\sum_{m\leq N}d(m) +O(N^{1-1/(\tau+1)+\varepsilon}), \end{align*} $$

where N is a sufficiently large integer, $\alpha $ is of finite type $\tau $ and $\beta \neq 0$. Previously, such estimates were only obtained for homogeneous Beatty sequences or for almost all $\alpha $.

Let $\mathbb {N}$ be the set of all nonnegative integers. For $S\subseteq \mathbb {N}$ and $n\in \mathbb {N}$, let $R_S(n)$ denote the number of solutions of the equation $n=s_1+s_2$, $s_1,s_2\in S$ and $s_1<s_2$. Let A be the set of all nonnegative integers which contain an even number of digits $1$ in their binary representations and $B=\mathbb {N}\setminus A$. Put $A_l=A\cap [0,2^l-1]$ and $B_l=B\cap [0,2^l-1]$. We prove that if $C \cup D=[0, m]\setminus \{r\}$ with $0<r<m$, $C \cap D=\emptyset $ and $0 \in C$, then $R_{C}(n)=R_{D}(n)$ for any nonnegative integer n if and only if there exists an integer $l \geq 1$ such that $m=2^{l}$, $r=2^{l-1}$, $C=A_{l-1} \cup (2^{l-1}+1+B_{l-1})$ and $D=B_{l-1} \cup (2^{l-1}+1+A_{l-1})$. Kiss and Sándor [‘Partitions of the set of nonnegative integers with the same representation functions’, Discrete Math. 340 (2017), 1154–1161] proved an analogous result when $C\cup D=[0,m]$, $0\in C$ and $C\cap D=\{r\}$.

For a set A of positive integers and any positive integer n, let $R_{1}(A, n)$, $R_{2}(A,n)$ and $R_{3}(A,n)$ denote the number of solutions of $a+a^{\prime }=n$ with $a, a^{\prime }\in A$ and the additional restriction that $a<a^{\prime }$ for $R_{2}$ and $a\leq a^{\prime }$ for $R_{3}$. We consider Problem 6 of Erdős et al. [‘On additive properties of general sequences’, Discrete Math. 136 (1994), 75–99] about locally small and locally large values of $R_{1}, R_{2}$ and $R_{3}$.

Let $k\geq 2$ be an integer. We prove that the 2-automatic sequence of odious numbers $\mathcal {O}$ is a k-additive uniqueness set for multiplicative functions: if a multiplicative function f satisfies a multivariate Cauchy’s functional equation $f(x_1+x_2+\cdots +x_k)=f(x_1)+f(x_2)+\cdots +f(x_k)$ for arbitrary $x_1,\ldots ,x_k\in \mathcal {O}$, then f is the identity function $f(n)=n$ for all $n\in \mathbb {N}$.

In 1946, Erdős and Niven proved that no two partial sums of the harmonic series can be equal. We present a generalisation of the Erdős–Niven theorem by showing that no two partial sums of the series $\sum _{k=0}^\infty {1}/{(a+bk)}$ can be equal, where a and b are positive integers. The proof of our result uses analytic and p-adic methods.

We study quantitative relationships between the triangle removal lemma and several of its variants. One such variant, which we call the triangle-free lemma, states that for each $\epsilon>0$ there exists M such that every triangle-free graph G has an $\epsilon$-approximate homomorphism to a triangle-free graph F on at most M vertices (here an $\epsilon$-approximate homomorphism is a map $V(G) \to V(F)$ where all but at most $\epsilon \left\lvert{V(G)}\right\rvert^2$ edges of G are mapped to edges of F). One consequence of our results is that the least possible M in the triangle-free lemma grows faster than exponential in any polynomial in $\epsilon^{-1}$. We also prove more general results for arbitrary graphs, as well as arithmetic analogues over finite fields, where the bounds are close to optimal.

We demonstrate that every difference set in a finite Abelian group is equivalent to a certain ‘regular’ covering of the lattice $ A_n = \{ \boldsymbol {x} \in \mathbb {Z} ^{n+1} : \sum _{i} x_i = 0 \} $ with balls of radius $ 2 $ under the $ \ell _1 $ metric (or, equivalently, a covering of the integer lattice $ \mathbb {Z} ^n $ with balls of radius $ 1 $ under a slightly different metric). For planar difference sets, the covering is also a packing, and therefore a tiling, of $ A_n $. This observation leads to a geometric reformulation of the prime power conjecture and of other statements involving Abelian difference sets.

For a polynomial progression

$$ \begin{align*}(x,\; x+P_1(y),\ldots,\; x+P_{t}(y)),\end{align*} $$

$P_1, \ldots ,\!P_t$

$$ \begin{align*}(x,\; x+y^2,\; x+2y^2,\; x+y^3,\; x+2y^3),\end{align*} $$

$\mathbb {Z}/N\mathbb {Z}$

$\mathbb {Z}/N\mathbb {Z}$