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).

中文翻译：

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$$ {\ mathbb {N}} $$ N的子集的指数密度和极限比

给定\（\ alpha，\ beta，\ gamma \ in [0,1] \）与\（\ alpha \ le \ beta \），我们证明存在\（{\ mathbb {N}} \ ），使其下指数密度和上指数密度以及下极限比率和上限比率分别等于\（\ alpha \），\（\ beta \），\（\ gamma \）和1。这个结果为Grekos等人提出的未解决问题提供了肯定的答案。（Unif Distrib Theory 6：117–130，2011）。