A subset $A$ of a finite abelian group $G$ is called $(k,l)$-sum-free if the sum of $k$ (not necessarily distinct) elements of $A$ never equals the sum of $l$ (not necessarily distinct) elements of $A$. We find an explicit formula for the maximum size of a $(k,l)$-sum-free subset in $G$ for all $k$ and $l$ in the case when $G$ is cyclic by proving that it suffices to consider $(k,l)$-sum-free intervals in subgroups of $G$. This simplifies and extends earlier results by Hamidoune and Plagne [‘A new critical pair theorem applied to sum-free sets in abelian groups’, Comment. Math. Helv. 79(1) (2004), 183–207] and Bajnok [‘On the maximum size of a $(k,l)$-sum-free subset of an abelian group’, Int. J. Number Theory 5(6) (2009), 953–971].

We establish finite analogues of the identities known as the Aoki–Ohno relation and the Le–Murakami relation in the theory of multiple zeta values. We use an explicit form of a generating series given by Aoki and Ohno.

Let $\mathbb{N}$ be the set of all nonnegative integers. For any set $A\subset \mathbb{N}$, let $R(A,n)$ denote the number of representations of $n$ as $n=a+a^{\prime }$ with $a,a^{\prime }\in A$. There is no partition $\mathbb{N}=A\cup B$ such that $R(A,n)=R(B,n)$ for all sufficiently large integers $n$. We prove that a partition $\mathbb{N}=A\cup B$ satisfies $|R(A,n)-R(B,n)|\leq 1$ for all nonnegative integers $n$ if and only if, for each nonnegative integer $m$, exactly one of $2m+1$ and $2m$ is in $A$.

Let $n$ be a positive integer. We obtain new Menon’s identities by using the actions of some subgroups of $(\mathbb{Z}/n\mathbb{Z})^{\times }$ on the set $\mathbb{Z}/n\mathbb{Z}$. In particular, let $p$ be an odd prime and let $\unicode[STIX]{x1D6FC}$ be a positive integer. If $H_{k}$ is a subgroup of $(\mathbb{Z}/p^{\unicode[STIX]{x1D6FC}}\mathbb{Z})^{\times }$ with index $k=p^{\unicode[STIX]{x1D6FD}}u$ such that $0\leqslant \unicode[STIX]{x1D6FD}<\unicode[STIX]{x1D6FC}$ and $u\mid p-1$, then

$$\begin{eqnarray}\mathop{\sum }_{x\in H_{k}}(x-1,p^{\unicode[STIX]{x1D6FC}})=\frac{\unicode[STIX]{x1D711}(p^{\unicode[STIX]{x1D6FC}})}{k}\bigg(1+k(\unicode[STIX]{x1D6FC}-\unicode[STIX]{x1D6FD})+u\frac{p^{\unicode[STIX]{x1D6FD}}-1}{p-1}\bigg),\end{eqnarray}$$

$\unicode[STIX]{x1D711}(n)$

For a given set $S\subset \mathbb{N}$, $R_{S}(n)$ is the number of solutions of the equation $n=s+s^{\prime },s<s^{\prime },s,s^{\prime }\in S$. Suppose that $m$ and $r$ are integers with $m>r\geq 0$ and that $A$ and $B$ are sets with $A\cup B=\mathbb{N}$ and $A\cap B=\{r+mk:k\in \mathbb{N}\}$. We prove that if $R_{A}(n)=R_{B}(n)$ for all positive integers $n$, then there exists an integer $l\geq 1$ such that $r=2^{2l}-1$ and $m=2^{2l+1}-1$. This solves a problem of Chen and Lev [‘Integer sets with identical representation functions’, Integers16 (2016), A36] under the condition $m>r$.

Let $\mathfrak{D}$ be a residually finite Dedekind domain and let $\mathfrak{n}$ be a nonzero ideal of $\mathfrak{D}$. We consider counting problems for the ideal chains in $\mathfrak{D}/\mathfrak{n}$. By using the Cauchy–Frobenius–Burnside lemma, we also obtain some further extensions of Menon’s identity.

We prove two conjectural congruences on the $(p-1)$th Apéry number, which were recently proposed by Z.-H. Sun.

Green and Tao famously proved in 2005 that any subset of the primes of fixed positive density contains arbitrarily long arithmetic progressions. Green had previously shown that, in fact, any subset of the primes of relative density tending to zero sufficiently slowly contains a three-term progression. This was followed by work of Helfgott and de Roton, and Naslund, who improved the bounds on the relative density in the case of three-term progressions. The aim of this note is to present an analogous result for longer progressions by combining a quantified version of the relative Szemerédi theorem given by Conlon, Fox and Zhao with Henriot's estimates of the enveloping sieve weights.

Let $F$ be an integral linear recurrence, $G$ an integer-valued polynomial splitting over the rationals and $h$ a positive integer. Also, let ${\mathcal{A}}_{F,G,h}$ be the set of all natural numbers $n$ such that $\gcd (F(n),G(n))=h$. We prove that ${\mathcal{A}}_{F,G,h}$ has a natural density. Moreover, assuming that $F$ is nondegenerate and $G$ has no fixed divisors, we show that the density of ${\mathcal{A}}_{F,G,1}$ is 0 if and only if ${\mathcal{A}}_{F,G,1}$ is finite.

Let $G$ be a finite abelian group, $A$ a nonempty subset of $G$ and $h\geq 2$ an integer. For $g\in G$, let $R_{A,h}(g)$ denote the number of solutions of the equation $x_{1}+\cdots +x_{h}=g$ with $x_{i}\in A$ for $1\leq i\leq h$. Kiss et al. [‘Groups, partitions and representation functions’, Publ. Math. Debrecen85(3) (2014), 425–433] proved that (a) if $R_{A,h}(g)=R_{G\setminus A,h}(g)$ for all $g\in G$, then $|G|=2|A|$, and (b) if $h$ is even and $|G|=2|A|$, then $R_{A,h}(g)=R_{G\setminus A,h}(g)$ for all $g\in G$. We prove that $R_{G\setminus A,h}(g)-(-1)^{h}R_{A,h}(g)$ does not depend on $g$. In particular, if $h$ is even and $R_{A,h}(g)=R_{G\setminus A,h}(g)$ for some $g\in G$, then $|G|=2|A|$. If $h>1$ is odd and $R_{A,h}(g)=R_{G\setminus A,h}(g)$ for all $g\in G$, then $R_{A,h}(g)=\frac{1}{2}|A|^{h-1}$ for all $g\in G$. If $h>1$ is odd and $|G|$ is even, then there exists a subset $A$ of $G$ with $|A|=\frac{1}{2}|G|$ such that $R_{A,h}(g)\not =R_{G\setminus A,h}(g)$ for all $g\in G$.

We consider a family of nonlinear rational recurrences of odd order which was introduced by Heideman and Hogan, and recently rediscovered in the theory of Laurent phenomenon algebras (a generalization of cluster algebras). All of these recurrences have the Laurent property, implying that for a particular choice of initial data (all initial values set to 1) they generate an integer sequence. For these particular sequences, Heideman and Hogan gave a direct proof of integrality by showing that the terms of the sequence also satisfy a linear recurrence relation with constant coefficients. Here we present an analogous result for the general solution of each of these recurrences.

Let ${\mathcal{A}}=\{a_{1}<a_{2}<\cdots \,\}$ be a set of nonnegative integers. Put $D({\mathcal{A}})=\gcd \{a_{k+1}-a_{k}:k=1,2,\ldots \}$ . The set ${\mathcal{A}}$ is an asymptotic basis if there exists $h$ such that every sufficiently large integer is a sum of at most $h$ (not necessarily distinct) elements of ${\mathcal{A}}$ . We prove that if the difference of consecutive integers of ${\mathcal{A}}$ is bounded, then ${\mathcal{A}}$ is an asymptotic basis if and only if there exists an integer $a\in {\mathcal{A}}$ such that $(a,D({\mathcal{A}}))=1$ .

We provide two new bounds on the number of visible points on exponential curves modulo a prime for all choices of primes. We also provide one new bound on the number of visible points on exponential curves modulo a prime for almost all primes.

An additive basis $A$ is finitely stable when the order of $A$ is equal to the order of $A\cup F$ for all finite subsets $F\subseteq \mathbb{N}$ . We give a sufficient condition for an additive basis to be finitely stable. In particular, we prove that $\mathbb{N}^{2}$ is finitely stable.

Let $P_{1},\ldots ,P_{k}:\mathbb{Z}\rightarrow \mathbb{Z}$ be polynomials of degree at most $d$ for some $d\geqslant 1$ , with the degree $d$ coefficients all distinct, and admissible in the sense that for every prime $p$ , there exists integers $n,m$ such that $n+P_{1}(m),\ldots ,n+P_{k}(m)$ are all not divisible by $p$ . We show that there exist infinitely many natural numbers $n,m$ such that $n+P_{1}(m),\ldots ,n+P_{k}(m)$ are simultaneously prime, generalizing a previous result of the authors, which was restricted to the special case $P_{1}(0)=\cdots =P_{k}(0)=0$ (though it allowed for the top degree coefficients to coincide). Furthermore, we obtain an asymptotic for the number of such prime pairs $n,m$ with $n\leqslant N$ and $m\leqslant M$ with $M$ slightly less than $N^{1/d}$ . This asymptotic is already new in general in the homogeneous case $P_{1}(0)=\cdots =P_{k}(0)=0$ . Our arguments rely on four ingredients. The first is a (slightly modified) generalized von Neumann theorem of the authors, reducing matters to controlling certain averaged local Gowers norms of (suitable normalizations of) the von Mangoldt function. The second is a more recent concatenation theorem of the authors, controlling these averaged local Gowers norms by global Gowers norms. The third ingredient is the work of Green and the authors on linear equations in primes, allowing one to compute these global Gowers norms for the normalized von Mangoldt functions. Finally, we use the Conlon–Fox–Zhao densification approach to the transference principle to combine the preceding three ingredients together. In the special case $P_{1}(0)=\cdots =P_{k}(0)=0$ , our methods also give infinitely many $n,m$ with $n+P_{1}(m),\ldots ,n+P_{k}(m)$ in a specified set primes of positive relative density $\unicode[STIX]{x1D6FF}$ , with $m$ bounded by $\log ^{L}n$ for some $L$ independent of the density $\unicode[STIX]{x1D6FF}$ . This improves slightly on a result from our previous paper, in which $L$ was allowed to depend on $\unicode[STIX]{x1D6FF}$ .

For a positive integer $n$ let

$$\begin{eqnarray}\mathfrak{P}_{n}=\mathop{\prod }_{\substack{ p \\ s_{p}(n)\geqslant p}}p,\end{eqnarray}$$

$p$

$s_{p}(n)$

$p$

$n$

$n$

$\mathfrak{P}_{n}$

$\mathfrak{P}_{n}$

$\sqrt{n}$

$n^{20/37}$

$\sqrt{n}$

$\unicode[STIX]{x1D705}\sqrt{n}/\text{log}\,n$

$\unicode[STIX]{x1D705}$

$\unicode[STIX]{x1D705}=2$

$\mathfrak{P}_{n}$

$\mathfrak{P}_{n+1}$

$\mathfrak{P}_{n}$

$n$

$\mathfrak{P}_{n}>\mathfrak{P}_{n+1}$

Let $A$ be a set of natural numbers. Recent work has suggested a strong link between the additive energy of $A$ (the number of solutions to $a_{1}+a_{2}=a_{3}+a_{4}$ with $a_{i}\in A$) and the metric Poissonian property, which is a fine-scale equidistribution property for dilates of $A$ modulo $1$. There appears to be reasonable evidence to speculate a sharp Khinchin-type threshold, that is, to speculate that the metric Poissonian property should be completely determined by whether or not a certain sum of additive energies is convergent or divergent. In this article, we primarily address the convergence theory, in other words the extent to which having a low additive energy forces a set to be metric Poissonian.

We prove that the exponent of distribution of $\unicode[STIX]{x1D70F}_{3}$ in arithmetic progressions can be as large as $\frac{1}{2}+\frac{1}{34}$, provided that the moduli is squarefree and has only sufficiently small prime factors. The tools involve arithmetic exponent pairs for algebraic trace functions, as well as a double $q$-analogue of the van der Corput method for smooth bilinear forms.

We provide lower bounds for $p$ -adic valuations of multisums of factorial ratios which satisfy an Apéry-like recurrence relation: these include Apéry, Domb and Franel numbers, the numbers of abelian squares over a finite alphabet, and constant terms of powers of certain Laurent polynomials. In particular, we prove Beukers’ conjectures on the $p$ -adic valuation of Apéry numbers. Furthermore, we give an effective criterion for a sequence of factorial ratios to satisfy the $p$ -Lucas property for almost all primes $p$ .

Let $a_{1},a_{2},\ldots ,a_{m}$ and $b_{1},b_{2},\ldots ,b_{l}$ be two sequences of pairwise distinct positive integers greater than $1$ . Assume also that none of the above numbers is a perfect power. If for each positive integer $n$ and prime number $p$ the number $\prod _{i=1}^{m}(1-a_{i}^{n})$ is divisible by $p$ if and only if the number $\prod _{j=1}^{l}(1-b_{j}^{n})$ is divisible by $p$ , then $m=l$ and $\{a_{1},a_{2},\ldots ,a_{m}\}=\{b_{1},b_{2},\ldots ,b_{l}\}$ .