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


11B83 Special sequences and polynomials
11K38 Irregularities of distribution, discrepancy
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[2] Rohrbach, H.; Volkmann, B., Verallgemeinerte asymptotische Dichten, J. Reine Angew. Math., 194, 195-209 (1955) · Zbl 0064.28003
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