# 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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