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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?

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