×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.