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!)

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.