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Bulletin of the London Mathematical Society Advance Access originally published online on March 12, 2008
Bulletin of the London Mathematical Society 2008 40(1):37-50; doi:10.1112/blms/bdm090
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© 2008 London Mathematical Society

p-adic subsets whose factorials satisfy a generalized Legendre formula

Sabine Evrard and Youssef Fares

Laboratoire de Mathématiques fondamentales et appliquées d’Amiens
CNRS UMR 6140
33 rue Saint Leu
80039 Amiens France
youssef.fares@u-picardie.fr

Received 23 February 2007.

Amice studied the notion of a regular compact subset in a local field K, with valuation v and maximal ideal M.In her work, she introduced the notion of well distributed sequences and showed that every regular compact subset S admits well distributed sequences and that its factorial sequence (n!S) satisfies a generalized Legendre formula:


Formula

for every integer n and where qi denotes the number of classes of S modulo Mi.

In this article, in more general settings, we show the converse assertions. More precisely, we prove that, for every precompact subset of any discrete valuation domain V, the following assertions are equivalent:

  1. the topological closure of S is a regular subset,
  2. S admits a very well distributed sequence,
  3. S satisfies the generalized Legendre formula.

To Hamadi Fares

2000 Mathematics Subject Classification (primary)11B65,11B50, (secondary)13F20.


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