If do not lie in the same -orbit, one can take a bump function supported in a small neighbourhood of and average it along the action to obtain a suitable function to separate . If instead lies in the orbit of , then the set is a non-trivial coset of some closed subgroup of . One can then locate a frequency which annihilates but not (because the characters on separate points, by Plancherel’s theorem), and then the modulated average will work if the support of is small enough.

]]>I am trying to understand the proof of Proposition 1. I got stuck in the first part of the proof, where you claim that we can assume the continuous function has the property that for all and , where is some non-zero character of the torus. I assume this is true because functions of this kind generate an algebra that separates points and hence, by Stone-Weierstrass (like you suggested in the proof), this algebra is dense in . However, I am stuck on proving that this family actually separates points. Can you please clarify the argument that I am missing here. Thank you very much!

]]>I see that I’m wrong. (Please delete this and the previous comment.)

]]>Thanks! I expected this question to be much more difficult – because it seems related to the complicated behavior of the (theta) function

,

near the irrational point on the unit circle – its natural boundary!

The function (defined e.g. in page 36, Entry 22(i) in Berndt's book "Ramanujan Notebooks" Part III, 1991)

From the proof of Entry 23(i) (page 38 in this book) it follows that

Where is the generating function for the partition function .

Therefore, behavior near the unit circle may be studied via the well known complicated behavior of there (in particular, it follows that has the unit circle as its natural boundary!)

]]>Ouawww I got it !!!! Thank you very much ! I love this trick.

]]>Yes.

Weyl’s theorem can be found for instance in Corollary 6 of my notes https://terrytao.wordpress.com/2010/03/28/254b-notes-1-equidistribution-of-polynomial-sequences-in-torii/ . The point is that when one expands out the square in the indicated sequence in terms of complex exponentials, one gets terms such as for various . For , the phase here contains an irrational nonconstant coefficient and so has asymptotic mean zero in n. If one worked with instead of then would instead be constant in and Weyl’s theorem would not apply.

]]>That’s why I’m asking the reference of the theorem.

]]>It is not clear why this argument is working for but not for .

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