Extension de quelques theoremes sur les densites de series d'elements de n a des series de sous-ensembles finis de n

M. Deza, Paul Erdös

Research output: Contribution to journalArticle

5 Citations (Scopus)

Abstract

For a sequence A = {Ak} of finite subsets of N we introduce: δ(A) = infm≤nA(m) 2n, d(A) = lim infn→∞ A(n) 2n, where A(m) is the number of subsets Ak ⊆ {1, 2, ..., m}. The collection of all subsets of {1, ..., n} together with the operation a ∪ b, (a ∩ b), (a * b = a ∪ b {minus 45 degree rule} a ∩ b) constitutes a finite semi-group N (semi-group N) (group N*). For N, N we prove analogues of the Erdös-Landau theorem: δ(A+B) ≥ δ(A)(1+(2λ)-1(1-δ(A>))), where B is a base of N of the average order λ. We prove for N, N, N* analogues of Schnirelmann's theorem (that δ(A) + δ(B) > 1 implies δ(A + B) = 1) and the inequalities λ ≤ 2h, where h is the order of the base. We introduce the concept of divisibility of subsets: a|b if b is a continuation of a. We prove an analog of the Davenport-Erdös theorem: if d(A) > 0, then there exists an infinite sequence {Akr}, where Akr

Original languageEnglish
Pages (from-to)295-308
Number of pages14
JournalDiscrete Mathematics
Volume12
Issue number4
DOIs
Publication statusPublished - 1975

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

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