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co.combinatorics – The captious exponent duty Answer

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co.combinatorics – The captious exponent duty

It is a identified truth [1] that, for each $cin (1,infty]$, it’s workable to discover a finite alphabet $mathcal{A}$ and a phrase $earn mathcal{A}^omega$ such that $w$ has captious exponent $c$. It appears unaffected to outline what I’d convene the captious exponent duty in two steps as follows.

Step 1

For each integer $nge2$, the $n$captious exponent duty $kappa_n$ is outlined by:
commence{equation}
xin[0,1];longmapsto ;kappa_n(x)=frac{1}{c_n(x)}
aim{equation}

the place $c_n(x)$ is the captious exponent of the $n$-base growth of $x$ (and it’s supposed $frac{1}{infty}=0$). Notice that $kappa_n$ just isn’t affected by the workable ambiguity within the growth of some rational factors (the captious exponent is $infty$ for each the workable expansions).

The meander of $kappa_n$ is $left[0,frac{4}{7}right]$ for $n=3$, $left[0,frac{5}{7}right]$ for $n=4$ and $left[0,frac{n-1}{n}right]$ when $n=2$ or $nge 5$. This is because of a outcome by Rao [2] protecting the final circumstances of a common surmise by Dejean on repetition thresholds for finite alphabets [3].

Step 2

The captious exponent duty $kappa$ is outlined by:
commence{equation}
kappa: xin[0,1];longmapsto; sup_{nge 2}{kappa_n(x)}in[0,1] aim{equation}

It is well seen that $kappa$ vanishes on completely regular actual numbers. Therefore, $kappa$ is Lebesgue-measurable and $int_0^1 kappa(x)dx=0$.

Looks love $kappa$ has a number of uncommon properties (it recollects loosely Conway’s base-13 duty). Almost each query about it I can cerebrate of appears non-trivial. I pose three of them.

Q1: Which Baire class (if any) does $kappa$ belong to?

Q2: Does $kappa$ have mounted and/or sporadic factors (other than the trifling mounted level 0)?

Q3: Does $kappa$ gain the worth 1?

[1]: Krieger, D., & Shallit, J. (2007). Every actual quantity larger than 1 is a captious exponent. Theoretical laptop science, 381(1-3), 177-182.

[2]: Rao, M. (2011). Last circumstances of Dejean’s surmise. Theoretical Computer Science, 412(27), 3010-3018.

[3]: Dejean, F. (1972). Sur un théorème de Thue. Journal of Combinatorial Theory, Series A, 13(1), 90-99.

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