The purpose of this paper is to investigate the properties of spectral and tiling subsets of cyclic groups, with an eye towards the spectral set conjecture [9] in one dimension, which states that a bounded measurable subset of $\mathbb {R}$ accepts an orthogonal basis of exponentials if and only if it tiles $\mathbb {R}$ by translations. This conjecture is strongly connected to its discrete counterpart, namely that, in every finite cyclic group, a subset is spectral if and only if it is a tile. The tools presented herein are refinements of recent ones used in the setting of cyclic groups; the structure of vanishing sums of roots of unity [20] is a prevalent notion throughout the text, as well as the structure of tiling subsets of integers [1]. We manage to prove the conjecture for cyclic groups of order $p^{m}q^{n}$ , when one of the exponents is $\leq 6$ or when $p^{m-2}<q^{4}$ , and also prove that a tiling subset of a cyclic group of order $p_{1}^{m}p_{2}\dotsm p_{n}$ is spectral.

The conjecture of Brown, Erdős and Sós from 1973 states that, for any k ≥ 3, if a 3-uniform hypergraph H with n vertices does not contain a set of k +3 vertices spanning at least k edges then it has o(n2) edges. The case k = 3 of this conjecture is the celebrated (6, 3)-theorem of Ruzsa and Szemerédi which implies Roth’s theorem on 3-term arithmetic progressions in dense sets of integers. Solymosi observed that, in order to prove the conjecture, one can assume that H consists of triples (a, b, ab) of some finite quasigroup Γ. Since this problem remains open for all k ≥ 4, he further proposed to study triple systems coming from finite groups. In this case he proved that the conjecture holds also for k = 4. Here we completely resolve the Brown–Erdős–Sós conjecture for all finite groups and values of k. Moreover, we prove that the hypergraphs coming from groups contain sets of size $\Theta (\sqrt k )$ which span k edges. This is best possible and goes far beyond the conjecture.

We give a new formula for the number of cyclic subgroups of a finite abelian group. This is based on Burnside’s lemma applied to the action of the power automorphism group. The resulting formula generalises Menon’s identity.

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].

The genus spectrum of a finite group G is the set of all g such that G acts faithfully on a compact Riemann surface of genus g. It is an open problem to find a general description of the genus spectrum of the groups in interesting classes, such as the Abelian p-groups. Motivated by earlier work of Talu for odd primes, we develop a general combinatorial method, for arbitrary primes, to obtain a structured description of the so-called reduced genus spectrum of Abelian p-groups, including the reduced minimum genus. In particular, we determine the complete genus spectrum for a large subclass, namely, those having ‘large’ defining invariants. With our method we construct infinitely many counterexamples to a conjecture of Talu, which states that an Abelian p-group is recoverable from its genus spectrum. Finally, we give a series of examples of our method, in the course of which we prove, for example, that almost all elementary Abelian p-groups are uniquely determined by their minimum genus, and that almost all Abelian p-groups of exponent p2 are uniquely determined by their minimum genus and Kulkarni invariant.

The main result in Y. O. Hamidoune's paper ‘Adding distinct congruence classes' (Combin. Probab. Comput.7 (1998) 81–87) is as follows. If S is a generating subset of a cyclic group G such that 0 ∉ S and |S| ⩾ 5, then the number of sums of the subsets of S is at least min(|G|, 2|S|). Unfortunately, the argument of the author, who, sadly, passed away in 2011, relies on a lemma whose proof is incorrect; in fact, the lemma is false for all cyclic groups of even order. In this short note we point out this mistake, correct the proof, and discuss why the main result is actually true for all finite abelian groups.

Let G be an additive abelian group, let n ⩾ 1 be an integer, let S be a sequence over G of length |S| ⩾ n + 1, and let ${\mathsf h}$(S) denote the maximum multiplicity of a term in S. Let Σn(S) denote the set consisting of all elements in G which can be expressed as the sum of terms from a subsequence of S having length n. In this paper, we prove that either ng ∈ Σn(S) for every term g in S whose multiplicity is at least ${\mathsf h}$(S) − 1 or |Σn(S)| ⩾ min{n + 1, |S| − n + | supp (S)| − 1}, where |supp(S)| denotes the number of distinct terms that occur in S. When G is finite cyclic and n = |G|, this confirms a conjecture of Y. O. Hamidoune from 2003.

The positive cohomology groups of a finite group acting on a ring vanish when the ring has a norm one element. In this note we give explicit homotopies on the level of cochains when the group is cyclic, which allows us to express any cocycle of a cyclic group as the coboundary of an explicit cochain. The formulas in this note are closely related to the effective problems considered in previous joint work with Eli Aljadeff.

The above-titled paper of mine appeared in the Bulletin of the London Mathematical Society, 32 (2000) 297–304. Regrettably, there is a careless error in the proofs of Theorems 6 and 8. In line 6 of the proof of Theorem 6, it is claimed that a certain subset must be a subgroup. For this to hold, the subset must contain the zero element. This need not be the case; the true deduction is that the subset is a coset, say $M+h$, of a subgroup $M$. Now $M$ and $M+h$ contain the same number of elements, and so the deduction that $M$ has $p$ elements is still correct.