Alain Plagne

Optimally small sumsets in finite abelian groups

Abstract Let G be a finite abelian group of order g. We determine, for all 1⩽r,s⩽g, the minimal size μG(r,s)=min|A+B| of sumsets A+B, where A and B range over all subsets of G of cardinality r and s, respectively. We do so by explicit construction. Our formula for μG(r,s) shows that this function only depends on the cardinality of G, not on its specific group structure. Earlier results on μG are recalled in the Introduction.

On large half-factorial sets in elementaryp-groups: Maximal cardinality and structural characterization

Half-factoriality is a central concept in the theory of non-unique factorization, with applications for instance in algebraic number theory. A subsetG0 of an abelian group is called half-factorial if the block monoid overG0, which is the monoid of all zero-sum sequences of elements ofG0, is a half-factorial monoid. In this paper we study half-factorial sets with large cardinality in elementaryp-groups. First, we determine the maximal cardinality of such half-factorial sets, and generalize a result which has been only known for groups of even rank. Second, we characterize the structure of all half-factorial sets with large cardinality (in a sense made precise in the paper). Both results have a direct application in the study of some counting functions related to factorization properties of algebraic integers.

Grekos’ S function has a linear growth

An exact additive asymptotic basis is a set of nonnegative integers such that there exists an integer h with the property that any sufficiently large integer can be written as a sum of exactly h elements of A. The minimal such h is the exact order of A (denoted by ord*(A)). Given any exact additive asymptotic basis A, we define A* to be the subset of A composed with the elements a E A such that A \ {a} is still an exact additive asymptotic basis. It is known that A \A* is finite. In this framework, a central quantity introduced by Grekos is the function S(h) defined as the following maximum (taken over all bases A of exact order h): S(h) = max limsup ord*(A \ {a}). In this paper, we introduce a new and simple method for the study of this function. We obtain a new estimate from above for S which improves drastically and in any case on all previously known estimates. Our estimate, namely S(h) h + 1. However, it is certainly not always optimal since S(2) = 3. Our last result shows that S(h) is in fact a strictly increasing sequence.

On the structure of subsets of an orderable group with some small doubling properties

Abstract The aim of this paper is to present a complete description of the structure of subsets S of an orderable group G satisfying | S 2 | = 3 | S | − 2 and 〈 S 〉 is non-abelian.

On the subgroup generated by a small doubling binary set

Let A be a subset of (Z/2Z)n, such that |2A| > 2|A|. In this paper, we prove that there exist a subgroup H of (Z/2Z)n and a subgroup P of H with |P| ≤ |H|/8 such that H contains 2A, and H\2A is either empty or a full P-coset. We use this result to obtain an upper bound for the cardinality of the subgroup 〈A〉 generated by A in terms of |A|. More precisely we show that if 0 ∈ A and |2A| = τ|A| then |〈A〉|/|A| is equal to τ if 1 ≤ τ > 7/4, and is less than 8τ/7 if 7/4 ≤ τ > 2. This result is optimal.

New lower bounds for covering codes

Abstract We develop two methods for obtaining new lower bounds for the cardinality of covering codes. Both are based on the notion of linear inequality of a code. Indeed, every linear inequality of a code (defined on F q n ) allows to obtain, using a classical formula (inequality (2) below), a lower bound on K q (n,R) , the minimum cardinality of a covering code with radius R . We first show how to get new linear inequalities (providing new lower bounds) from old ones. Then, we prove some formulae that improve on the classical formula (2) for linear inequalities of some given types. Applying both methods to all the classical cases of the literature, we improve on nearly 20% of the best lower bounds on K q (n,R) .

Small doubling in ordered groups: Generators and structure

We prove several new results on the structure of the subgroup generated by a small doubling subset of an ordered group, abelian or not. We obtain precise results generalizing Freimans 3k-3 and 3k-2 theorems in the integers and several further generalizations.

THE DAVENPORT CONSTANT OF A BOX

Given an additively written abelian group

A New Upper Bound for B22] Sets

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Yahya Ould Hamidoune's mathematical journey

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