A point to set principle for finite-state dimension

Effective dimension was introduced in [7, 6] as an effectivization of Hausdorff dimension. One of its generalizations is finite-state dimension [2] that is a robust notion that interacts with compression and characterizes Borel normality [1].

In [8]  Lutz and Lutz proved a point-to-set principle that characterizes Hausdorff dimension in terms of relativized effective dimension. This principle has already produced a number of interesting results in geometric fractal theory through computability based proofs (see [5, 9] and a number of more recent results such as [4]).

In this paper we provide a characterization of finite-state dimension on Euclidean space based on the finite-state information content of a real number at a certain precision, which also provides an alternative characterization of Borel normality. This new characterization gives rise to a natural and robust relativization of finite state dimension with the strong property of a finite-state dimension point-to-set principle. We finish with open questions on the equidistribution properties of the corresponding relativized normality.

Let $\Sigma$ be a finite alphabet. We write $\Sigma^{*}$ for the set of all (finite) strings over $\Sigma$ and $\Sigma^{\infty}$ for the set of all (infinite) sequences over $\Sigma$. We write $|x|$ for the length of a string or sequence $x$, and we write $\lambda$ for the empty string, the string of length 0. For $x\in\Sigma^{*}\cup\Sigma^{\infty}$ and $0\leq n<|x|$, we write $x\upharpoonright n=x[0..n-1]$. For $w\in\Sigma^{*}$ and $x\in\Sigma^{*}\cup\Sigma^{\infty}$, we say that $w$ is a prefix of $x$, and we write $w\sqsubseteq x$, if $x\upharpoonright|w|=w$.

A $\Sigma$ finite-state transducer ($\Sigma$-FST) is a 4-tuple $T=(Q,\delta,\nu,q_{0})$, where

• •

  $Q$ is a nonempty, finite set of states,

• •

  ${\delta}:{Q\times\Sigma}\to{Q}$ is the transition function,

• •

  ${\nu}:{Q\times\Sigma}\to{\Sigma^{*}}$ is the output function, and

• •

  $q_{0}\in Q$ is the initial state.

For $q\in Q$ and $w\in\Sigma^{*}$, we define the output from state $q$ on input $w$ to be the string $\nu(q,w)$ defined by the recursion

	$\displaystyle\nu(q,\lambda)=\lambda,$
	$\displaystyle\nu(q,wa)=\nu(q,w)\nu(\delta(q,w),a)$

for all $w\in\Sigma^{*}$ and $a\in\Sigma$. We then define the output of $T$ on input $w\in\Sigma^{*}$ to be the string $T(w)=\nu(q_{0},w)$.

For each integer $b\geq 1$ we let $\Sigma_{b}=\{0,1,\ldots,b-1\}$ be the alphabet of base-$b$ digits. We use infinite sequences over $\Sigma_{b}$ to represent real numbers in $[0,1)$. Each $S\in\Sigma_{b}^{\infty}$ is associated the real number $\operatorname{real}_{b}(S)=\sum\limits_{i=1}^{\infty}S[i-1]b^{-i}$ and for each $x\in[0,1)$, $\operatorname{seq}_{b}(x)$ is the infinite sequence $S$ that does not finish with infinitely many $b-1$ and such that $x=\operatorname{real}_{b}(S)$.

A set of real numbers $A\subseteq[0,1)$ is represented by the set

$$\operatorname{seq}_{b}(A)=\{\operatorname{seq}_{b}(\alpha)\mid\alpha\in A\}$$

of sequences. If $X\subseteq\Sigma_{b}^{\infty}$ then

$$\operatorname{real}_{b}(X)=\{\operatorname{real}_{b}(x)\mid x\in X\}.$$

We will denote with ${\mathcal{D}}_{b}$ the set of rational numbers that have finite representation in base $b$, that is,

$${\mathcal{D}}_{b}=\left\{q\left|\,\operatorname{seq}_{b}(q)=w0^{\infty},w\in\Sigma_{b}^{*}\right.\right\}.$$

We will write $\operatorname{real}_{b}(w)$ for $\operatorname{real}_{b}(w0^{\infty})$ when $w\in\Sigma_{b}^{*}$.

Finite-state dimension was introduced in [2] on the space of infinite sequences over a finite alphabet. The original definition in terms of gambling was proven robust by several characterizations in terms of information lossless compression [2] and several versions of entropy [1].

Here we present an alternative definition on the Euclidean space and then prove its equivalence with [2].

We next define the finite-state dimension of points and sets.

The definition of finite-state dimension from [2] is usually done in a space of infinite sequences, while identifying $[0,1)$ and $\Sigma_{b}^{\infty}$ through $\operatorname{seq}_{b}$ or base-$b$ representation.

Doty and Moser [3] proved that finite dimension on sequences can be characterized in terms of finite-state transducers.

We next show that the notion of information content at a certain precision characterizes finite-state dimension.

For each $w\in\Sigma_{b}^{*}\cup\Sigma_{b}^{\infty}$, let $comp(w)$ be the complementary of $w$, that is, $comp(w)[i]=b-1-w[i]$ for $0\leq i<|w|$.

To prove this claim, notice that $\mathrm{dim}_{\mathrm{FS}}^{b}(x)$ needs to be witnessed either by approximations from above or for approximations from below and that tighter approximations can be delayed.

That is, for every FST $T$ there exist and infinitely many $n_{i}$ such that

$$\lim_{i}\frac{{\mathrm{K}}^{T}_{b^{-n_{i}}}(x)}{n_{i}}=\liminf_{n}\frac{{\mathrm{K}}^{T}_{b^{-n}}(x)}{n}.$$

For each $i$ let $w_{i}$ be such that ${\mathrm{K}}^{T}(w_{i})={\mathrm{K}}^{T}_{b^{-n_{i}}}(x)$ and $|x-\operatorname{real}_{b}(w_{i})|<b^{-n_{i}}$. Let $m_{i}$ be such that $b^{-m_{i}-1}\leq|x-\operatorname{real}_{b}(w_{i})|<b^{-m_{i}}\leq b^{-n_{i}}$. Then $\frac{{\mathrm{K}}^{T}_{b^{-m_{i}}}(x)}{m_{i}}\leq\frac{{\mathrm{K}}^{T}_{b^{-n_{i}}}(x)}{n_{i}}$, and for $T^{\prime}(\pi)=comp(T(\pi))$, either

$$\liminf_{n}\frac{{\mathrm{K}}^{T}(S\upharpoonright n)}{n}\leq\lim_{i}\frac{{\mathrm{K}}^{T}_{b^{-m_{i}}}(x)}{m_{i}}$$

or

$$\liminf_{n}\frac{{\mathrm{K}}^{T^{\prime}}(comp(S)\upharpoonright n)}{n}\leq\lim_{i}\frac{{\mathrm{K}}^{T}_{b^{-m_{i}}}(x)}{m_{i}}.$$

Therefore either $\mathrm{dim}_{\mathrm{FS}}(S)\leq\mathrm{dim}_{\mathrm{FS}}^{b}(x)$ or $\mathrm{dim}_{\mathrm{FS}}(comp(S))\leq\mathrm{dim}_{\mathrm{FS}}^{b}(x)$ and the claim follows.

$\Box$

Since finite-state dimension in the space of sequences characterizes Borel normality [1], we have an alternative characterization of normality in terms of finite-state dimension in the Euclidean space.

We denote as separator a set $S\subseteq[0,1)$ such that $S$ is countable and dense in [0,1].

For each separator enumerator $f$ we can define information content in $[0,1)$ relative to $f$.

We can generalize Borel normality through the same relativization.

Given this natural relativization of finite-state dimension we next prove a point-to-set principle stating that for every set $A$ there exists an SE $f$ such that classical Hausdorff dimension of $A$ is exactly $f$-finite-state dimension. This implies that classical geometrical measure theory results can be obtained using only finite-state dimension.

Notice that the previous theorem holds even when fixing a particular countable dense set. In terms of Borel normality, it shows that reordering the set ${\mathcal{D}}_{b}$ of base-$b$ finite representation numbers is enough to obtain normality for any other base.

We expect that our main theorem will prove new lower bounds on Hausdorff dimension in different settings. Notice that the result can be directly translated into any separable metric space and any reasonable gauge family.

Our result helps clarify the oracle role in the point to set principles. The next step would be to classify the different enumerations of a countable dense set.

We believe that the notion of $f$-normal sequence can be of independent interest with robustness properties inherited from those of the original concept, for instance from the fact that $x$ is $b$-normal exactly when the sequence $(b^{n}x)_{n}$ is equidistributed modulo 1.