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I have an easier to follow argument for lemma 1, which I'll add when I have some time to log on my pc. Also, the simple case generalizes to arbitrary submersions, which I'll also add.
Bjorn - it still gets a little messy choosing the binary digits correctly. Your answer seems pretty good to me and, as it's way past my bedtime now, I'll leave it at that.
Ok, cool. I was about to write something similar as per my comment above, using the binary expansion of theta/pi using the idea behind the construction of the Liouville constant. Think it boils down to the same basic idea as this answer though.
