## 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

Zbl 0041.18301
Full Text:

### 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
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.