Diophantine equation in the integer quantum hall effect

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Hi everybody!

I'm studying the IQHE and I want to understand the rise of the diophantine equation. I read the thouless article but it was no so conclusive.
I've also read the kohmoto article (phys rev B 1989) and he says that that property comes from the darboux theorem but i don't understand why… I've investigated the magnetic translation group but now I cannot do the link with the diophantine equation.

Can you help me?

Thanks a lot!

alex
 

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See also the reference they cite for the derivation using magnetic translation group arguement.
 

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