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 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.

A set A of positive integers is a Bh-set if all the sums of the form a1 + . . . + ah, with a1,. . .,ah ∈ A and a1 ⩽ . . . ⩽ ah, are distinct. We provide asymptotic bounds for the number of Bh-sets of a given cardinality contained in the interval [n] = {1,. . .,n}. As a consequence of our results, we address a problem of Cameron and Erdős (1990) in the context of Bh-sets. We also use these results to estimate the maximum size of a Bh-sets contained in a typical (random) subset of [n] with a given cardinality.

We show that for any coprime integers λ1, . . ., λk and any finite A ⊂ $\mathbb{Z}$, one has

|\lambda_1 \cdot A + \cdots + \lambda_k \cdot A| \geq (|\lambda_1| + \cdots + |\lambda_k|)|A|- C,