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Assume that $a>0$, $b_0>0$, $b_1>0$ and that $-2<b_1-b_0\leq -1$. Define $f_0(x) = \lfloor b_0-ax \rfloor$ and $f_1(x) = \lceil b_1-ax \rceil$. Finally, define the following set:

$$\mathcal{D} = \Big\{n\in\{0,1,2,...\}: f_0(n)=f_1(n) \Big\}$$

Is it possible to analytically characterize $\mathcal{D}$?

A Few Additional Notes

  1. If it is significantly easier to analytically characterize $$\Big\{n\in\{0,1,2,...\}: f_0(n)<f_1(n) \Big\} \text{ or } \Big\{n\in\{0,1,2,...\}: f_0(n)>f_1(n) \Big\}$$ please feel free to replace $\mathcal{D}$ with one of the above sets. (I would appreciate your answer just as much! Which is a lot; thank you so much for taking the time to even read this.)
  2. Apologies for the question title being more general than needed. (Kept hitting the word limit!)
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