4 arxiv tag and ergodic tag
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Suppose you take a $\pm 1$ sequence and you want to "improve it" by taking pointwise limits of translates. What properties can you guarantee to get in the limit?

Two examples illustrate what I think should be the extremes. If your sequence is random, then every finite subsequence occurs infinitely often, which means that it is easy to find translates that converge pointwise to any sequence you like. In particular, you can make the final sequence constant, though what interests me is that it is highly structured.

By contrast, let's suppose that the sequence is quasiperiodic, or more concretely that you take a real number $\alpha$ and define $x_n$ to be 1 or -1 according to whether the integer part of $\alpha n$ is even or odd. In that case, it is not hard to prove that every pointwise limit is quasiperiodic as well. (Unless I've made a mistake, the result is that there must be some $\beta$ such that $z_n$ is 1 or -1 according to whether the integer part of $\alpha n+\beta$ is even or odd.)

What can be said in general? Can we always "get rid of the randomness" and find a highly structured limit? And what does "highly structured" mean? Perhaps that an associated dynamical system is compact, though I'd ideally like a characterization in terms of the sequence itself.

This question ought to be meat and drink to ergodic theorists. Indeed, I feel slightly guilty for not knowing the answer already. It arises naturally in a Polymath project (which can be found by searching for "Erdos discrepancy problem").

Added later: I've just realized one simple lemma. Suppose you want to get rid of some finite subsequence. Then you can do it if you can find arbitrarily long subsequences that avoid that finite subsequence. So either you can get rid of the subsequence, or it occurs in the original sequence with bounded gaps. I think that means you can reach a sequence such that every finite subsequence that appears appears with bounded gaps.

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Suppose you take a $\pm 1$ sequence and you want to "improve it" by taking pointwise limits of translates. What properties can you guarantee to get in the limit?

Two examples illustrate what I think should be the extremes. If your sequence is random, then every finite subsequence occurs infinitely often, which means that it is easy to find translates that converge pointwise to any sequence you like. In particular, you can make the final sequence constant, though what interests me is that it is highly structured.

By contrast, let's suppose that the sequence is quasiperiodic, or more concretely that you take a real number $\alpha$ and define $x_n$ to be 1 or -1 according to whether the integer part of $\alpha n$ is even or odd. In that case, it is not hard to prove that every pointwise limit is quasiperiodic as well. (Unless I've made a mistake, the result is that there must be some $latex \beta$ \beta$such that$z_n$is 1 or -1 according to whether the integer part of$\alpha n+\beta\$ is even or odd.)

What can be said in general? Can we always "get rid of the randomness" and find a highly structured limit? And what does "highly structured" mean? Perhaps that an associated dynamical system is compact, though I'd ideally like a characterization in terms of the sequence itself.

This question ought to be meat and drink to ergodic theorists. Indeed, I feel slightly guilty for not knowing the answer already. It arises naturally in a Polymath project (which can be found by searching for "Erdos discrepancy problem").

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