Let $A\subset \mathbb{N}$. Its upper and lower asymptotic densities are defined by $$\overline{d}(A)=\limsup\limits_{n\to \infty}\frac{\#(A\cap \{1,2,\ldots,n\})}{n}$$ and $$\underline{d}(A)=\liminf\limits_{n\to \infty}\frac{\#(A\cap \{1,2,\ldots,n\})}{n}$$ respectively, where $\#(B)$ denotes the cardinality of $B$. It is well known that for any $0\le s\le t\le 1$, there exists a set $S\subset \mathbb{N}$ such that $\underline{d}(S)=s$ and $\overline{d}(S)=t$ (see, for example, (2.10) and (2.11) in the paper https://www.worldscientific.com/doi/epdf/10.1142/S1793042117500051).
Now I encounter the following density which can be regarded as an invariant of the upper asymptotic density. For $0<\theta\le 1$, define
$$\overline{d}_\theta(A)=\limsup\limits_{n\to \infty}\inf_{k\in \{n,n+1,\ldots, [\frac{n}{\theta}]\}}\frac{\#(A\cap \{1,2,\ldots,k\})}{k},$$ where $[x]$denotes the integer part of $x$. Clearly, $\overline{d}_\theta(A)=\overline{d}(A)$ when $\theta=1$.
My question is: Fix $0<\theta\le 1$. For any $0\le s\le t\le 1$, does there exists a set $S\subset \mathbb{N}$ such that $\underline{d}(S)=s$ and $\overline{d}_\theta(S)=t$?