Additive Decompositions, Random Allocations, and Threshold Phenomena

  • Authors:

  • Olivier Dubois;Guy Louchard;Jacques Mandler

  • Affiliations:

  • LIP6, C.N.R.S.-Université Paris 6, 4 place Jussieu, 75252 Paris cedex 05, France (e-mail: Olivier.Dubois@lip6.fr, Jacques.Mandler@lip6.fr);Université Libre de Bruxelles, Département d'Informatique, CP 212, Boulevard du Triomphe, B-1050 Bruxelles, Belgium (e-mail: louchard@ulb.ac.be.);LIP6, C.N.R.S.-Université Paris 6, 4 place Jussieu, 75252 Paris cedex 05, France (e-mail: Olivier.Dubois@lip6.fr, Jacques.Mandler@lip6.fr)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2004

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Abstract

An additive decomposition of a set $I$ of nonnegative integers is an expression of $I$ as the arithmetic sum of two other such sets. If the smaller of these has $p$ elements, we have a $p$-decomposition. If $I$ is obtained by randomly removing $n^{\alpha}$ integers from $\{0,\dots,n-1\}$, decomposability translates into a balls-and-urns problem, which we start to investigate (for large $n$) by first showing that the number of $p$-decompositions exhibits a threshold phenomenon as $\alpha$ crosses a $p$-dependent critical value. We then study in detail the distribution of the number of 2-decompositions. For this last case we show that the threshold is sharp and we establish the threshold function.