How do irrational numbers give incommensurate potential (in lattice models)?

I am trying to understand Aubry-Andre model. It has the following form $$H = \sum_n c_n^\dagger c_{n+1}+H.C.+V\sum_n \cos{(2\pi\beta n)}c_n^\dagger c_n$$

This reference (at 3rd page) says that if $$\beta$$ is irrational (rational) then period of potential is quasi-periodic incommensurate (periodic commensurate) with underlying lattice period.

Question 1: What does incommensurate potential mean here?

Question 2: How does irrational $$\beta$$ guarantee that potential is quasi-period incommensurate with underlying lattice?

Further more, this reference says that with irrational $$\beta$$ (they are taking inverse of Golden mean i.e. $$(\sqrt 5 - 1)/2)$$ to avoid the unwanted boundary effects, we have to take system of size of any number from Fibonacci series.

Question 3: How does system of size of any Fibonacci series' number avoid unwanted boundary effects?

• Your pdf link broken; it links to a file named untitled.pdf on someone's desktop. Mar 25, 2019 at 0:14
• @DavidHammen I am sorry. I have updated it. Thank you for pointing it out. Mar 25, 2019 at 16:28