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Let $\mu$ be a probability measure on the complex unit circle $\mathbb{T}$. Are the following two assertions equivalent?

  1. $\limsup_{n\to\infty}|\hat{\mu}(n)|=1$.
  2. There exists an increasing sequence ${n_k}$ such that $\lim_{k\rightarrow\infty}\hat{\mu}(n_k)=1$.
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You don't want absolute values in the second limit? – Alex R. Jan 31 '13 at 18:16
That's the point. – user25640 Jan 31 '13 at 18:45
Using Lebesgue decomposition theorem, we just have to deal with discrete measures. – Davide Giraudo Feb 4 '13 at 10:14

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