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Simple question (understable at a pre-graduated level) :

Given a sequence $(x_n)$ taking values in the interval $[0,1]$, is it always possible to construct a "symmetrizing" sequence $(s(n))$ of $0$ or $1$, such that the sequence $$ y_n \doteq (-1)^{s(n)}x_n + s(n) $$ is "symmetric in $[0,1]$" in the sense that for any $ 0 \leq a < b < \frac{1}{2} $, the sequence $$ \text{Card} ( n \leq N : y_n \in [a,b] ) - \text{Card} ( n \leq N : y_n \in [1-b,1-a]) $$ is bounded as $N$ increases ?

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    $\begingroup$ Hi! Unfortunately questions on undergraduate mathematics are not on-topic on MathOverflow, which has a deliberately narrow scope (roughly, PhD-level and above). Were you thinking of other StackExchange site for mathematics questions? Best of luck with your studies! $\endgroup$
    – David Roberts ♦
    Commented Aug 17 at 12:42
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    $\begingroup$ @DavidRoberts I have the impression that "simple question at a pre-graduate level" just means it has a simple formulation by elementary notions, but not necessarily that it has a simple answer at pre-graduate level (like e.g. the FLThm). $\endgroup$
    – Pietro Majer
    Commented Aug 17 at 14:24
  • $\begingroup$ Equivalently, does there exists a $c$ such that we can colour any finite set of points red and blue such that in every axis-aligned rectangle the number of red points and the number of blue points differs by at most $c$? $\endgroup$
    – 1001
    Commented Aug 17 at 20:07
  • $\begingroup$ Something to do with discrepancy $\endgroup$
    – 1001
    Commented Aug 17 at 20:19
  • $\begingroup$ @PietroMajer in that case, it's fine. Someone had flagged a previous comment that made the point much less nicely and I wanted to replace that with an informative and cheerful version. $\endgroup$
    – David Roberts ♦
    Commented Aug 18 at 2:41

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No.

Take $x_n \in [0,1/2]$. The problem reduces to the online interval discrepancy problem: https://en.wikipedia.org/wiki/Geometric_discrepancy

For the reduction, colour the point $x_n$ black if $s_n = 1$ and white if $s_n = 0$.

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