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On densities and gaps. (English) Zbl 0622.10044
The authors investigate certain relations between the asymptotic densities of subsets of a set $$A=\{m_ 1,m_ 2,...\}$$ of positive integers and the lengths of gaps in A by means of the ”analytical” properties of the two-dimensional set $$S(A)=\{(\underline d(B),\bar d(B))\in {\mathbb{R}}^ 2$$; $$B\subseteq A\}$$, where ḏ(B) and $$\bar d(B)$$ denote the lower and upper asymptotic density of B.
A special role here is played by the function $$f_ S$$, whose graph is the lower part of the boundary of S(A). So for instance, the right derivatives $$D^+_ f(0)$$ of f at the origin is not less than $$\lambda (A)=\limsup_{j\to \infty}m_{j+1}/m_ j$$. Conversely, given a set $$S\subseteq {\mathbb{R}}^ 2$$ satisfying certain conditions and a real number $$\lambda$$ such that $$1\leq \lambda \leq D^+_ f(0)$$, where $$f=f_ S$$ then there exists a set A of positive integers such that $$S(A)=S$$ and $$\lambda (A)=\lambda$$. They also give a characterization of $$D^+_ f(0)$$ of the function f corresponding to a given set A in terms of the lengths of intervals in which the set A possesses ”few” elements, etc.
Reviewer: Št.Porubský

##### MSC:
 11B83 Special sequences and polynomials 11K38 Irregularities of distribution, discrepancy
##### Keywords:
density set; asymptotic density; gap density
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##### References:
 [1] Grekos, G, Répartition des densités des sous-suites d’une suite d’entiers, J. number theory, 10, 177-191, (1978) · Zbl 0388.10033 [2] Rohrbach, H; Volkmann, B, Verallgemeinerte asymptotische dichten, J. reine angew. math., 194, 195-209, (1955) · Zbl 0064.28003
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