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Is it true that a sequence of real numbers $\{a_n\}$ converges to zero if and only if the sequences $\{\sin^2(nh)a_n\}$ $(h \in \mathbb{R})$ all converge to zero?

In case the answer is affirmative (and please skip the obvious implication):

1. How far can we replace the condition "for every value of $h \in \mathbb{R}$" by a (much) weaker condition "for every value of $h \in A \subset \mathbb{R}$"? Will, for example, some countable or finite set $A$ suffice?

2. Is there anything special with the function $\sin^2(x)$ or can we replace it with (many) other functions?

The topmost question arose from a strategy to prove the following little nut:

If $f : \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function such that the function $\Delta_h f(x) := f(x + h) - f(x)\ $ is smooth for every value of $h$, then $f$ is itself smooth.

This problem is in turn related to this previous MO problem.

Edit June 2, 2014. A colleague just "reminded me" that the OP is a consequence of the

Cantor-Lebesgue Theorem. Let $A_n(x) := c_n e^{inx} + c_{-n}e^{-inx}$. If $A_n(x)$ tends to zero as $n$ tends to infinity for all $x$ in some set of positive measure, then $c_n$ and $c_{-n}$ tend to zero as $n$ tends to infinity.

See this paper by R.L. Cooke for a nice presentation.

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The following theorem about uniform distribution on the circle $\mathbb T^1$ is well known (and is due to Weyl): given an increasing sequence of positive integers $k_n$, there is $z\in\mathbb T^1$ such that $z^{k_n}$ is uniformly distributed in $\mathbb T^1$ (using the complex notation). In fact, this statement is true if you replace $\mathbb T^1$ by any compact connected second countable abelian group and almost every $z$ is good in the sense of Haar measure.

Now to your question. Assume that $a_n$ does not converge to zero. Then it has a subsequence $a_{k_n}$ which is bounded away from zero. By the theorem mentioned above there is $h\in\mathbb R$ with $\sin^2(hk_n)$ running densely through $[0,1]$. Consequently, $\sin^2(hk_n)a_{k_n}$ has a subsequence bounded away from zero and so it does not converge to zero.

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  • $\begingroup$ By the way, what is special about $\sin$ is that it corresponds to the projection of $\mathbb T^1$ onto $[-1,1]$ as a part of the imaginary axis and that it takes dense subsets of $\mathbb T^1$ onto dense subsets of $[-1,1]$. This should give an idea on how to search for other functions. $\endgroup$ – William of Baskerville May 30 '14 at 13:19
  • $\begingroup$ I checked this in the book ``Uniform distribution of sequences'' by Kuipers and Niederreiter. The circle case is due to Weyl (Theorem 4.1, page 32). The group case is due to Hartman and Ryll-Nardzevski (at the bottom of page 279, the same book). Can you help me out what the Ryll-Nardzevski fixed point theorem says? $\endgroup$ – William of Baskerville May 30 '14 at 13:51
  • $\begingroup$ Do you have a reference for the theorem you cite? There is a very famous fixed-point theorem by the same name that seems to skew google-searches. $\endgroup$ – alvarezpaiva May 30 '14 at 13:52
  • $\begingroup$ The fixed-point theorem says that a group of isometries preserving a weakly compact convex subset of a Banach space must have a fixed point in the convex set. $\endgroup$ – alvarezpaiva May 30 '14 at 13:56
  • $\begingroup$ I just checked the reference to the original paper : Ryll-Nardzewski, C. (1962). "Generalized random ergodic theorems and weakly almost periodic functions". Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 10: 271–275. Probably, the results are related ... $\endgroup$ – alvarezpaiva May 30 '14 at 13:57

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