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Given    $$\alpha ,\beta ,\gamma \in [0,1]$$    α  ,  β  ,  γ  ∈  [  0  ,  1  ]      with    $$\alpha \le \beta $$    α  ≤  β     , we prove that there exists a subset of    $${\mathbb {N}}$$   N     such that its lower and upper exponential densities and its lower and upper limit ratios are equal to    $$\alpha $$   α    ,    $$\beta $$   β    ,    $$\gamma $$   γ     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).

On Exponential Densities and Limit Ratios of Subsets of $${\mathbb {N}}$$