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On weighted densities. (English) Zbl 1195.11018

Let \(A\) be a set of positive integers. The authors study, in detail, a special class of densities: the lower \(\alpha\)-density \(\underline{d}_\alpha(A)\) and the upper \(\alpha\)-density \(\overline{d}^\alpha(A)\) defined as the \(\liminf\) and \(\limsup\) of the \(\frac{\sum_{n\leq N, n\in A}n^\alpha}{\sum_{n\leq N}n^\alpha}\), respectively, where \(N\to\infty\). For example, if \(\alpha=0\) then we have asymptotic densities and if \(\alpha=-1\), we have logarithmic densities.
Motivated by monotonicity of \(\underline{d}_\alpha(A)\) and \(\overline{d}^\alpha(A)\) with respect to \(\alpha\) which was proved by C. T . Rajagopal [Am. J. Math. 70, 157–166 (1948; Zbl 0041.18301)], the authors prove, for arbitrary \(A\), that \(\underline{d}_\alpha(A)\) and \(\overline{d}^\alpha(A)\) are continuous in \(\alpha\in(-1,\infty)\).
As the main result they prove the continuity of \(\underline{d}_\alpha(A)\) and \(\overline{d}^\alpha(A)\) in \(\alpha=-1\), for a class of sets \(A\) of all integer numbers from intervals (\(a_n,b_n]\), \(a_n<b_n<a_{n+1}\), \(\log b_n-\log b_{n-1}\leq B\), \(\log b_n-\log a_n\geq C>0\) for \(n=1,2,\dots\), where \(C,B\) are constants. The given proof is long but elementary.

MSC:

11B05 Density, gaps, topology
11P70 Inverse problems of additive number theory, including sumsets

Citations:

Zbl 0041.18301
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References:

[1] A. Fuchs and R. Antonini Giuliano: Théorie générale des densités. Rend. Acc. Naz. delle Scienze detta dei XL, Mem. di Matematica, 108 (XIV, 14) (1990), 253–294.
[2] H. Halberstam and K. F. Roth: Sequences. Oxford Univ. Press, 1966.
[3] G. H. Hardy and M. Riesz: The General Theory of Dirichlet Series. Cambridge Univ. Press, 1952.
[4] L. Mišík: Sets of positive integers with prescribed values of densities. Mathematica Slovaca 52 (2002), 289–296. · Zbl 1005.11004
[5] L. Mišík and J. T. Tóth: Logarithmic density of a sequence of integers and density of its ratio set. Journal de Théorie des Nombres de Bordeaux 15 (2003), 309–318. · Zbl 1130.11304
[6] C. T. Rajagopal: Some limit theorems. Amer. J. Math. 70 (1948), 157–166. · Zbl 0041.18301 · doi:10.2307/2371942
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