I am looking for a function with the following property:

Let $v_1,v_2$ be two linearly independent vectors in $\mathbb{R}^2.$

I am given a smooth function $g:(0,1) \rightarrow (0,\infty).$

I am trying to understand if there exists a smooth (non-constant) function $f:(0,1) \times \mathbb R^2 \rightarrow \mathbb R^2$ with the property that for all $n,m \in \mathbb Z$ such that $g(t)=n/m$, where $n/m$ is a reduced fraction, we have for all $x \in \mathbb R^2$

$$f(t,mv_1+x)=f(t,x)=f(t,nv_2+x)$$ and $n,m$ are the minimal periods of the function $f$, i.e. for all natural numbers $1\le n_0 <n$ we do not have

$$f(t,nv_2+x)=f(t,x)$$ and the same for $m.$