Mediterranean Journal of Mathematics ( IF 1.216 ) Pub Date : 2020-06-28 , DOI: 10.1007/s00009-020-01559-7
J. Li, L. Olsen

Given $$\alpha ,\beta ,\gamma \in [0,1]$$ with $$\alpha \le \beta$$, we prove that there exists a subset of $${\mathbb {N}}$$ such that its lower and upper exponential densities and its lower and upper limit ratios are equal to $$\alpha$$, $$\beta$$, $$\gamma$$ and 1, respectively. This result provides an affirmative answer to an open problem posed by Grekos et al. (Unif Distrib Theory 6:117–130, 2011).

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