Let v(k) be the smallest integer larger than 1 that does not occur among the denominators in any identity of the form

\begin{align*}1=\frac1{n_1}+\cdots+\frac1{n_k},\end{align*}

$1 \leq n_1\lt\cdots\lt n_k$

$v(k)\gg k!$

$c\gt0$

\begin{align*}v(k)\ge e^{c k^2}\end{align*}

Let $r\ge 1$ be an integer. For any multiple index $\mathbf {s}=(s_1,s_2,\ldots ,s_r) \in \mathbb {Z}_{\geq 1}^r$ with $s_r > 1$, the multiple zeta value (MZV for short) is defined by

$$ \begin{align*} \zeta(s_1,s_2,\ldots,s_r):=\sum_{1\leq k_1<k_2<\cdots<k_r} \frac{1}{k_1^{s_1}k_2^{s_2}\cdots k_r^{s_r}} \end{align*} $$

and the multiple t-value is defined by

$$ \begin{align*} t(s_1,s_2,\ldots,s_r):=\sum_{1\leq k_1<k_2<\cdots<k_r} \frac{1}{(2k_1-1)^{s_1}(2k_2-1)^{s_2}\cdots(2k_r-1)^{s_r}}. \end{align*} $$

If the index is empty, then we define the value $t(\emptyset ):=1$. We denote by $\{a_1,a_2,\ldots ,a_k\}^d$ the sequence formed by repeating the sequence $\{a_1,a_2,\ldots ,a_k\}$ exactly d times. Let $H(a,b)=\zeta (\{2\}^a,3,\{2\}^b)$ and $T(a,b):=t(\{2\}^a,3,\{2\}^b)$. By using the Lai–Lupu–Orr integral expressions for $H(a,b)$ and $T(a,b)$, and the properties of the Beta and Gamma functions, we show that for any nonnegative integers a and b,

$$ \begin{align*} H(a,b):=\frac{-4\pi^{2a+2b+2}}{(2a+2)!}\sum_{n=0}^{\infty} \frac{\zeta(2n)}{(2n+2a+2)(2n+2a+3)\cdots(2n+2a+2b+3)2^{2n}} \end{align*} $$

and

$$ \begin{align*} T(a,b)=\frac{-2}{(2a+1)!}\bigg(\frac{\pi}{2}\bigg)^{2a+2b+2} \sum_{n=0}^{\infty}\frac{\zeta(2n)}{(2n+2a+1)(2n+2a+2)\cdots(2n+2a+2b+2)2^{2n}}. \end{align*} $$

This confirms two conjectures proposed by Lupu [‘Another look at Zagier’s formula for multiple zeta values involving Hoffman elements’, Math. Z. 301 (2022), 3127–3140].

Our goal is to provide simple and practical algorithms in higher-order Fourier analysis which are based on spectral decompositions of operators. We propose a general framework for such algorithms and provide a detailed analysis of the quadratic case. Our results reveal new spectral aspects of the theory underlying higher-order Fourier analysis. Along these lines, we prove new inverse and regularity theorems for the Gowers norms based on higher-order character decompositions. Using these results, we prove a spectral inverse theorem and a spectral regularity theorem in quadratic Fourier analysis.

An $S_k$-set in a group $\Gamma $ is a set $A\subseteq \Gamma $ such that $\alpha _1\dots \alpha _k=\beta _1\dots \beta _k$ with $\alpha _i,\beta _i\in A$ implies $(\alpha _1,\ldots ,\alpha _k)=(\beta _1,\ldots ,\beta _k)$. An $S_k'$-set is a set such that $\alpha _1\beta _1^{-1}\dots \alpha _k\beta _k^{-1}=1$ implies that there exists i such that $\alpha _i=\beta _i\text { or }\beta _i=\alpha _{i+1}$. We give explicit constructions of large $S_k$-sets in the groups $\mathrm {Sym}(n)$ and $\mathrm {Alt}(n)$ and $S_2$-sets in $\mathrm {Sym}(n)\times \mathrm {Sym}(n)$ and $\mathrm {Alt}(n)\times \mathrm {Alt}(n)$. We give probabilistic constructions which yield large $S_2'$-sets in $\mathrm {Sym}(n)$. We also give upper bounds on the size of $S_k$-sets in certain groups, improving the trivial bound by a constant multiplicative factor. We describe some connections between $S_k$-sets and extremal graph theory. In particular, we determine up to a constant factor the minimum outdegree of a digraph which guarantees even cycles with certain orientations. As applications, we improve the upper bound on Hamilton paths which pairwise create a two-part cycle of given length, and we show that a directed version of the Erdős–Simonovits compactness conjecture is false.

We study a pair of finite multisets of nonnegative integers which have coinciding representation functions. We show that there is a close connection to factorisation of polynomials into cyclotomic polynomials. As an application, we give a complete characterisation of pairs of such sets whose union is the interval $[0,m]$ with two elements removed.

Let A be a subset of an additive abelian semigroup S and let $hA$ be the h-fold sumset of A. The following question is considered. Let $(A_q)_{q=1}^{\infty }$ be a strictly decreasing sequence of sets in S and let $A = \bigcap _{q=1}^{\infty } A_q$. When does one have

$$ \begin{align*} hA = \bigcap_{q=1}^{\infty} hA_q \end{align*} $$

for some or all $h \geq 2$?

Let $\triangle $ denote the integers represented by the quadratic form $x^2+3y^2$ and denote the numbers represented as a sum of two squares. For a nonzero integer a, let be the set of integers n such that $n \in \triangle $ and . We conduct a census of in short intervals by showing that there exists a constant $H_{a}> 0$ with

for large x. To derive this result and its generalisation, we use a theorem of Tolev [‘On the remainder term in the circle problem in an arithmetic progression’, Proc. Steklov Inst. Math. 276 (2012), 261–274].

Recently, Harrington et al. [‘Every arithmetic progression contains infinitely many b-Niven numbers’, Bull. Aust. Math. Soc. 109(3) (2024), 409–413] proved that every arithmetic progression contains infinitely many base-b Niven numbers for any fixed $b\ge 2$. We use a sparse repunit construction to treat a structured two-base version of the same problem, showing that every arithmetic progression with common difference relatively prime to b contains infinitely many integers that are simultaneously b-Niven and $b^k$-Niven (indeed, we can obtain simultaneous $b^\ell $-Niven-ness for $\ell =1,\ldots , k$).

If A and B are subsets of an abelian group, their sumset is $A+B:=\{a+b:a\in A, b\in B\}$. We study sumsets in discrete abelian groups, where at least one summand has positive upper Banach density.

Jin proved in [27] that if A and B are sets of integers having positive upper Banach density, then $A+B$ is piecewise syndetic. Bergelson, Furstenberg, and Weiss [4] improved the conclusion to “$A+B$ is piecewise Bohr.” In [2] this was shown to be qualitatively optimal, in the sense that if $C\subseteq \mathbb Z$ is piecewise Bohr, then there are $A, B\subseteq \mathbb Z$ having positive upper Banach density such that $A+B\subseteq C$.

We improve these results by establishing a strong correspondence between sumsets in discrete abelian groups, level sets of convolutions in compact abelian groups, and sumsets in compact abelian groups. Our proofs avoid measure preserving dynamics and nonstandard analysis, and our results apply to discrete abelian groups of any cardinality.

In a previous work, Bettin, Koukoulopoulos, and Sanna prove that if two sets of natural numbers A and B have natural density 1, then their product set $A \cdot B \;:\!=\; \{ab \;:\; a \in A, b \in B\}$ also has natural density 1. They also provide an effective rate and pose the question of determining the optimal rate. We make progress on this question by constructing a set A of density 1 such that $A\cdot A$ has a “large” complement.

Gowers and Hatami initiated the inverse theory for the uniformity norms $U^k$ of matrix-valued functions on non-abelian groups by proving a 1%-inverse theorem for the $U^2$-norm and relating it to stability questions for almost representations. In this paper, we take a step toward an inverse theory for higher-order uniformity norms of matrix-valued functions on arbitrary groups by examining the 99% regime for the $U^k$-norm on perfect groups of bounded commutator width.

This analysis prompts a classification of Leibman’s quadratic maps between non-abelian groups. Our principal contribution is a complete description of these maps via an explicit universal construction. From this classification we deduce several applications: A full classification of quadratic maps on arbitrary abelian groups; a proof that no nontrivial polynomial maps of degree greater than one exist on perfect groups; stability results for approximate polynomial maps.

We prove two results related to the solutions of norm form equations. First, we give a finiteness result for sums of terms of linear recurrence sequences appearing in the coordinates of solutions of norm form equations. Second, we give a finiteness result concerning solutions of norm form equations representable as sums of S-units with a fixed number of terms. To prove these results, we use a deep result concerning the finiteness of the solutions of polynomial-exponential equations and S-unit equations.

We use spectral theory and algebraic geometry to establish a higher-degree analogue of a Szemerédi–Trotter-type theorem over finite fields, with an application to polynomial expansion.

We compute the Galois groups of the reductions modulo a prime number p of the generating series of Apéry numbers, Domb numbers and Almkvist–Zudilin numbers. We observe in particular that their behaviour is governed by congruence conditions on p.

Let $\mathbb {N}$ be the set of all nonnegative integers. For a set $A\subseteq \mathbb {N}$, let $R_2(A,n)$ and $R_3(A,n)$ be the number of solutions of the equation $n=a_1+a_2$ with $a_1<a_2, a_1,a_2\in A$ and with $a_1\le a_2, a_1,a_2\in A$, respectively. If $-N\le g\le N$, Yan [‘On the structure of partition which the difference of their representation function is a constant’, Period. Math. Hungar. 82 (2021), 149–152] showed that there is a set $A\subseteq \mathbb {N}$ such that $R_i(A,n)-R_i(\mathbb {N}\setminus A,n)=g$ for all integers $n\ge 2N-1$, where N is a positive integer. In this paper, we prove that if $g_1,g_2$ are nonnegative integers with $g_1\neq g_2$, then there does not exist $A\subseteq \mathbb {N}$ such that $R_i(A,2n)-R_i(\mathbb {N}\setminus A,2n)=g_1$ and $R_i(A,2n+1)-R_i(\mathbb {N}\setminus A,2n+1)=g_2$ for all sufficiently large integers n.

We show that for any set $A\subset {\mathbb N}$ with positive upper density and any $\ell ,m \in {\mathbb N}$, there exist an infinite set $B\subset {\mathbb N}$ and some $t\in {\mathbb N}$ so that $\{mb_1 + \ell b_2 \colon b_1,b_2\in B\ \text {and}\ b_1<b_2 \}+t \subset A,$ verifying a conjecture of Kra, Moreira, Richter and Robertson. We also consider the patterns $\{mb_1 + \ell b_2 \colon b_1,b_2\in B\ \text {and}\ b_1 \leq b_2 \}$, for infinite $B\subset {\mathbb N}$ and prove that any set $A\subset {\mathbb N}$ with lower density $\underline {\!\mathrm {d}}(A)>1/2$ contains such configurations up to a shift. We show that the value $1/2$ is optimal and obtain analogous results for values of upper density and when no shift is allowed.

We introduce a relaxation of stability, called almost sure stability, which is insensitive to perturbations by subsets of Loeb measure $0$ in a non-standard finite group. We show that almost sure stability satisfies a stationarity principle in the sense of geometric stability theory for measure-independent elements. We apply this principle to deduce the existence of squares in dense almost surely stable subsets of Cartesian products of non-standard finite groups, possibly non-abelian. Our results imply qualitative asymptotic versions for Cartesian products of finite groups. In the final section, we establish the existence of $3\times 2$-grids (and thus of L-shapes) in dense almost surely stable two-dimensional subsets of finite abelian groups of odd order.

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)$.