Partitioning ℝ into sets $A$ and $B$, such that the measures of $A$ and $B$ in each non-empty open interval have an "almost" non-zero constant ratio

Motivation: I want to partition $\mathbb{R}$ into sets $A$ and $B$, where the measures of $A$ and $B$ in each non-empty open interval have positive ratios with an upper and lower bound, such that the bounds have the smallest absolute difference.

In other words, I want to partition $\mathbb{R}$ into sets $A$ and $B$, such that the measures of $A$ and $B$ in each non-empty open interval have an "almost" non-zero constant ratio.

Thereby, suppose $\lambda$ is the Lebesgue measure on the Borel $\sigma$-algebra: i.e., $\mathcal{B}(\mathbb{R})$

Question:

Does there exist an example of sets $A,B\subset\mathbb{R}$ and lower bound $\mathfrak{c}$, where:

1. $A\cup B=\mathbb{R}$
2. $A\cap B=\emptyset$
3. for all non-empty open intervals $I:=(a,b)\subset\mathbb{R}$, such that $c:\mathbb{R}^{2}\times\mathcal{B}(\mathbb{R})^{2}\to\mathbb{R}$$$\lambda(A\cap I)=c(a,b,A,B)\cdot\lambda(B \cap I)$$ where for constant $q:\mathbb{R}^{2}\times\mathcal{B}(\mathbb{R})^{2}\to\mathbb{R}$ and $r:\mathbb{R}^{2}\times\mathcal{B}(\mathbb{R})^{2}\to\mathbb{R}$$$q(a,b,A,B)\le c(a,b,A,B)\le r(a,b,A,B)$$such that lower bound $\mathfrak{c}$ satisfies: $$\mathfrak{c}\le r(a,b,A,B)-q(a,b,A,B)$$ where we want $A,B\in\mathcal{B}(\mathbb{R})$, such that: $$\mathfrak{c}=r(a,b,A,B)-q(a,b,A,B)?$$