Quelques problèmes additifs : bases, pseudo-puissances et ensembles k-libres

Abstract : Widely studied in N or Z, we are interested in additive bases in infinite abelian groups. We get some results about the functions E, X and S, which caracterize the behaviour of a basis when we remove an element. We also study the set A of pseudo s-th powers, which is an additive basis of order s+1. We wonder what is the minimal size of an additive complement of sA, that is a set B such that sA+B contains all large enough integers. With respect to this problem, we prove quite precise theorems which are tantamount to asserting that a threshold phenomenon occurs. Finally, we establish the maximal size of a k-free set in Z/nZ. The study of this quantity strongly depends on the arithmetical relative properties of n and k. That is why we use different methods depending on cases. In particular, we show a result on combinatorial trees for the general case.
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Contributor : Victor Lambert <>
Submitted on : Thursday, July 9, 2015 - 2:20:07 PM
Last modification on : Wednesday, March 27, 2019 - 4:10:22 PM
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  • HAL Id : tel-01174654, version 1

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Victor Lambert. Quelques problèmes additifs : bases, pseudo-puissances et ensembles k-libres. Mathématiques [math]. Ecole Polytechnique, 2015. Français. ⟨tel-01174654⟩

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