Timeline for Radial limit does not exist almost everywhere

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May 26, 2015 at 2:33	history	edited	GH from MO	CC BY-SA 3.0	deleted 26 characters in body
May 26, 2015 at 2:04	history	edited	GH from MO	CC BY-SA 3.0	deleted 413 characters in body
May 26, 2015 at 1:06	comment	added	GH from MO		@Lucia: I had similar ideas initially, but then I abandoned them. For some reason I thought that $h$ must grow sufficiently fast with $N$. I am now convinced that your idea works, and I suggest that you give it as an answer (so that it can be voted for and accepted officially).
May 26, 2015 at 1:02	comment	added	Lucia		Note that almost every real number $\theta$ will have the property that its binary expansion will have arbitrarily long strings of zeros. Now suppose $N$ is such that $\Vert 2^N \theta \Vert \le 2^{-h}$, and consider $f(e^{-1/2^N} e^{2\pi i\theta})$ and $f(e^{-1/2^{N+h}} e^{2\pi i \theta})$. They should differ by $\gg h$, showing that radial limits don't exist.
May 26, 2015 at 1:02	history	edited	GH from MO	CC BY-SA 3.0	added 8 characters in body
May 26, 2015 at 0:54	history	edited	GH from MO	CC BY-SA 3.0	added 125 characters in body
May 26, 2015 at 0:54	comment	added	Lev Borisov		Sorry, posted a comment by mistake -- have no idea how to touch this.
May 26, 2015 at 0:52	history	edited	GH from MO	CC BY-SA 3.0	added 125 characters in body
May 26, 2015 at 0:39	history	edited	GH from MO	CC BY-SA 3.0	added 345 characters in body
May 26, 2015 at 0:07	history	edited	GH from MO	CC BY-SA 3.0	added 13 characters in body
May 25, 2015 at 22:47	comment	added	Erika L		Thanks. The Tauberian theorem might be involved and actually Theorem 6.4 of Zygmund seems to be very similar to the high-indices theorem, which directly implies the stated result. However I feel like these theorems are an over-kill and there might be a way to prove it just using results in Stein's book.
May 25, 2015 at 22:08	history	answered	GH from MO	CC BY-SA 3.0