# Dimension Spectra of Lines

## Abstract

This paper investigates the algorithmic dimension spectra of lines in the Euclidean plane. Given any line *L* with slope *a* and vertical intercept *b*, the dimension spectrum \({{\mathrm{sp}}}(L)\) is the set of all effective Hausdorff dimensions of individual points on *L*. We draw on Kolmogorov complexity and geometrical arguments to show that if the effective Hausdorff dimension \(\dim (a, b)\) is equal to the effective packing dimension \({{\mathrm{Dim}}}(a, b)\), then \({{\mathrm{sp}}}(L)\) contains a unit interval. We also show that, if the dimension \(\dim (a, b)\) is at least one, then \({{\mathrm{sp}}}(L)\) is infinite. Together with previous work, this implies that the dimension spectrum of any line is infinite.

## 1 Introduction

Algorithmic dimensions refine notions of algorithmic randomness to quantify the density of algorithmic information of individual points in continuous spaces. The most well-studied algorithmic dimensions for a point \(x\in \mathbb {R}^n\) are the *effective Hausdorff dimension*, \(\dim (x)\), and its dual, the *effective packing dimension*, \({{\mathrm{Dim}}}(x)\) [1, 7]. These dimensions are both algorithmically and geometrically meaningful [3]. In particular, the quantities \(\sup _{x\in E}\dim (x)\) and \(\sup _{x\in E}{{\mathrm{Dim}}}(x)\) are closely related to classical Hausdorff and packing dimensions of a set \(E\subseteq \mathbb {R}^n\) [5, 8], and this relationship has been used to prove nontrivial results in classical fractal geometry using algorithmic information theory [8, 10, 12].

*(effective Hausdorff) dimension spectrum*of a set \(E \subseteq \mathbb {R}^n\), i.e., the set

*a*and vertical intercept

*b*, we ask what \({{\mathrm{sp}}}(L_{a,b})\) might be. It was shown by Turetsky that, for every \(n\ge 2\), the set of all points in \(\mathbb {R}^n\) with effective Hausdorff 1 is connected, guaranteeing that \(1\in {{\mathrm{sp}}}(L_{a,b})\). In recent work [10], we showed that the dimension spectrum of a line in \(\mathbb {R}^2\) cannot be a singleton. By proving a general lower bound on \(\dim (x,ax+b)\), which is presented as Theorem 5 here, we demonstrated that

*any*line has infinite cardinality.

We begin by reviewing definitions and properties of algorithmic information in Euclidean spaces in Sect. 2. In Sect. 3, we sketch our technical approach and state our main technical lemmas. In Sect. 4 we prove our first main theorem and state our second main theorem. We conclude in Sect. 5 with a brief discussion of future directions.

## 2 Preliminaries

### 2.1 Kolmogorov Complexity in Discrete Domains

*conditional Kolmogorov complexity*of binary string \(\sigma \in \{0,1\}^*\) given a binary string \(\tau \in \{0,1\}^*\) is the length of the shortest program \(\pi \) that will output \(\sigma \) given \(\tau \) as input. Formally, it is

*U*is a fixed universal prefix-free Turing machine and \(\ell (\pi )\) is the length of \(\pi \). Any \(\pi \) that achieves this minimum is said to

*testify*to, or be a

*witness*to, the value \(K(\sigma |\tau )\). The

*Kolmogorov complexity*of a binary string \(\sigma \) is \(K(\sigma )=K(\sigma |\lambda )\), where \(\lambda \) is the empty string. These definitions extend naturally to other finite data objects, e.g., vectors in \(\mathbb {Q}^n\), via standard binary encodings; see [6] for details.

### 2.2 Kolmogorov Complexity in Euclidean Spaces

*Kolmogorov complexity*of a point \(x\in \mathbb {R}^m\) at

*precision*\(r\in \mathbb {N}\) is the length of the shortest program \(\pi \) that outputs a

*precision*-

*r*rational estimate for

*x*. Formally, it is

*x*. The

*conditional Kolmogorov complexity of*

*x*

*at precision*

*r*

*given*\(y\in \mathbb {R}^n\)

*at precision*\(s\in \mathbb {R}^n\) is

*r*and

*s*are equal, we abbreviate \(K_{r,r}(x|y)\) by \(K_r(x|y)\). As the following lemma shows, these quantities obey a chain rule and are only linearly sensitive to their precision parameters.

### Lemma 1

As a matter of notational convenience, if we are given a nonintegral positive real as a precision parameter, we will always round up to the next integer. For example, \(K_{r}(x)\) denotes \(K_{\lceil r\rceil }(x)\) whenever \(r\in (0,\infty )\).

### 2.3 Effective Hausdorff and Packing Dimensions

*gales*, which generalize martingales. Subsequently, Athreya, et al., defined effective packing dimension, also using gales [1]. Mayordomo showed that effective Hausdorff dimension can be characterized using Kolmogorov complexity [11], and Mayordomo and J. Lutz showed that effective packing dimension can also be characterized in this way [9]. In this paper, we use these characterizations as definitions. The

*effective Hausdorff dimension*and

*effective packing dimension*of a point \(x\in \mathbb {R}^n\) are

*x*. Guided by the information-theoretic nature of these characterizations, J. Lutz and N. Lutz [8] defined the

*lower*and

*upper conditional dimension*of \(x\in \mathbb {R}^m\) given \(y\in \mathbb {R}^n\) as

### 2.4 Relative Complexity and Dimensions

By letting the underlying fixed prefix-free Turing machine *U* be a universal *oracle* machine, we may *relativize* the definition in this section to an arbitrary oracle set \(A \subseteq \mathbb {N}\). The definitions of \(K^A(\sigma |\tau )\), \(K^A(\sigma )\), \(K^A_r(x)\), \(K^A_r(x|y)\), \(\dim ^A(x)\), \({{\mathrm{Dim}}}^A(x)\)\(\dim ^A(x|y)\), and \({{\mathrm{Dim}}}^A(x|y)\) are then all identical to their unrelativized versions, except that *U* is given oracle access to *A*.

We will frequently consider the complexity of a point \(x \in \mathbb {R}^n\)*relative to a point*\(y \in \mathbb {R}^m\), i.e., relative to a set \(A_y\) that encodes the binary expansion of *y* is a standard way. We then write \(K^y_r(x)\) for \(K^{A_y}_r(x)\). J. Lutz and N. Lutz showed that \(K_r^y(x)\le K_{r,t}(x|y)+K(t)+O(1)\) [8].

## 3 Background and Approach

In this section we describe the basic ideas behind our investigation of dimension spectra of lines. We briefly discuss some of our earlier work on this subject, and we present two technical lemmas needed for the proof our main theorems.

The dimension of a point on a line in \(\mathbb {R}^2\) has the following trivial bound.

### Observation 2

For all \(a,b,x\in \mathbb {R}\), \(\dim (x,ax+b)\le \dim (x,a,b)\).

*x*for which the approximate converse

Specifically, for every \(s\in [0,1]\), we want to find an *x* of effective Hausdorff dimension *s* such that (1) holds. Note that equality in Observation 2 implies (1).

### Observation 3

This observation suggests an approach, whenever \(\dim ^{a,b}(x)>\dim (a,b)\), for showing that \(\dim (x,ax+b)\ge \dim (x,a,b)\). Since (*a*, *b*) is, in this case, the unique low-dimensional pair such that \((x,ax+b)\) lies on \(L_{a,b}\), one might naïvely hope to use this fact to derive an estimate of (*x*, *a*, *b*) from an estimate of \((x,ax+b)\). Unfortunately, the dimension of a point is not even semicomputable, so algorithmically distinguishing (*a*, *b*) requires a more refined statement.

### 3.1 Previous Work

The following lemma, which is essentially geometrical, is such a statement.

### Lemma 4

**(N. Lutz and Stull**[10]

**).**Let \(a,b,x\in \mathbb {R}\). For all \((u,v)\in \mathbb {R}^2\) such that \(u x+v=ax+b\) and \(t=-\log \Vert (a,b)-(u,v)\Vert \in (0,r]\),

Roughly, if \(\dim (a,b)<\dim ^{a,b}(x)\), then Lemma 4 tells us that \(K_r(u,v)>K_r(a,b)\) unless (*u*, *v*) is very close to (*a*, *b*). As \(K_r(u,v)\) is upper semicomputable, this is algorithmically useful: We can enumerate all pairs (*u*, *v*) whose precision-*r* complexity falls below a certain threshold. If one of these pairs satisfies, approximately, \(ux+v=ax+b\), then we know that (*u*, *v*) is close to (*a*, *b*). Thus, an estimate for \((x,ax+b)\) algorithmically yields an estimate for (*x*, *a*, *b*).

In our previous work [10], we used an argument of this type to prove a general lower bound on the dimension of points on lines in \(\mathbb {R}^2\):

### Theorem 5

**(N. Lutz and Stull**[10]

**).**For all \(a,b,x\in \mathbb {R}\),

The strategy in that work is to use oracles to artificially lower \(K_r(a,b)\) when necessary, to essentially force \(\dim (a,b)<\dim ^{a,b}(x)\). This enables the above argument structure to be used, but lowering the complexity of (*a*, *b*) also weakens the conclusion, leading to the minimum in Theorem 5.

### 3.2 Technical Lemmas

In the present work, we circumvent this limitation and achieve inequality (1) by controlling the choice of *x* and placing a condition on (*a*, *b*). Adapting the above argument to the case where \(\dim (a,b)>\dim ^{a,b}(x)\) requires refining the techniques of [10]. In particular, we use the following two technical lemmas, which strengthen results from that work. Lemma 6 weakens the conditions needed to compute an estimate of (*x*, *a*, *b*) from an estimate of \((x,ax+b)\).

### Lemma 6

Let \(a,b,x\in \mathbb {R}\), \(k \in \mathbb {N}\), and \(r_0=1\). Suppose that \(r_1,\ldots , r_k\in \mathbb {N}\), \(\delta \in \mathbb {R}_+\), and \(\varepsilon ,\eta \in \mathbb {Q}_+\) satisfy the following conditions for every \(1\le i\le k\).

- 1.
\(r_i \ge \log (2|a|+|x|+6)+r_{i-1}\).

- 2.
\(K_{r_i}(a,b)\le \left( \eta +\varepsilon \right) r_i\).

- 3.
For every \((u,v)\in \mathbb {R}^2\) such that \(t=-\log \Vert (a,b)-(u,v)\Vert \in (r_{i-1},r_i]\) and \(ux+v=ax+b\), \(K_{r_i}(u,v)\ge \left( \eta -\varepsilon \right) r_i+\delta \cdot (r_i- t)\).

Lemma 7 strengthens the oracle construction of [10], allowing us to control complexity at multiple levels of precision.

### Lemma 7

- 1.
For every \(t \le r_1\), \(K^D_t(z) =\min \{\eta r_1,K_t(z)\}+ O(\log r_k)\)

- 2.For every \(1 \le i \le k\),$$\begin{aligned} K^D_{r_i}(z) = \eta r_1 + \sum _{j =2}^i \min \{\eta (r_j - r_{j-1}), K_{r_j, r_{j-1}}(z \, | \, z)\} + O(\log r_k)\,. \end{aligned}$$
- 3.
For every \(t\in \mathbb {N}\) and \(x\in \mathbb {R}\), \(K^{z,D}_t(x) = K^z_t(x) + O(\log r_k)\).

## 4 Main Theorems

We are now prepared to prove our two main theorems. We first show that, for lines \(L_{a, b}\) such that \(\dim (a, b) = {{\mathrm{Dim}}}(a, b)\), the dimension spectrum \({{\mathrm{sp}}}(L_{a,b})\) contains the unit interval.

### Theorem 8

Let \(a, b \in \mathbb {R}\) satisfy \(\dim (a, b) = {{\mathrm{Dim}}}(a, b)\). Then for every \(s \in [0, 1]\) there is a point \(x\in \mathbb {R}\) such that \(\dim (x, ax + b) = s + \min \{\dim (a,b), 1\}\).

### Proof

*a*,

*b*). That is, there is some constant \(c \in \mathbb {N}\) such that for all \(r \in \mathbb {N}\), \(K^{a, b}_r(x) \ge r - c\). By Theorem 5,

*a*,

*b*). Define the sequence of natural numbers \(\{h_j\}_{j \in \mathbb {N}}\) inductively as follows. Define \(h_0 = 1\). For every \(j > 0\), let

*x*[

*r*] is the

*r*th bit of

*x*. Define \(x \in \mathbb {R}\) to be the real number with this binary expansion. Then \(K_{sh_j}(x)=sh_j+O(\log sh_j)\).

*k*is bounded by a constant depending only on

*s*and \(\eta \). Therefore a \(o(r_k) = o(r_i)\) for all \(r_i\). Let \(D_{r} = D(r_1,\ldots , r_k, (a, b), \eta )\) be the oracle defined in Lemma 7. We first note that, since \(\dim (a, b) = {{\mathrm{Dim}}}(a, b)\),

*j*. Hence, condition 2 of Lemma 6 is satisfied.

*x*,

*j*.

*x*,

*j*. Hence the conditions of Lemma 6 are satisfied, and we have

*x*,

*j*. Hence,

### Theorem 9

Let \(a, b \in \mathbb {R}\) such that \(\dim (a, b) \ge 1\). Then for every \(s \in [\frac{1}{2}, 1]\) there is a point \(x\in \mathbb {R}\) such that \(\dim (x, ax + b) \in \left[ \frac{3}{2} + s - \frac{1}{2s}, s + 1\right] \).

### Corollary 10

Let \(L_{a, b}\) be any line in \(\mathbb {R}^2\). Then the dimension spectrum \({{\mathrm{sp}}}(L_{a, b})\) is infinite.

### Proof

Let \((a, b) \in R^2\). If \(\dim (a, b) < 1\), then by Theorem 5 and Observation 2, the spectrum \({{\mathrm{sp}}}(L_{a, b})\) contains the interval \([\dim (a, b), 1]\). Assume that \(\dim (a, b) \ge 1\). By Theorem 9, for every \(s \in [\frac{1}{2}, 1]\), there is a point *x* such that \(\dim (x, ax + b) \in [\frac{3}{2} + s - \frac{1}{2s}, s + 1]\). Since these intervals are disjoint for \(s_n = \frac{2n - 1}{2n}\), the dimension spectrum \({{\mathrm{sp}}}(L_{a, b})\) is infinite.

## 5 Future Directions

We have made progress in the broader program of describing the dimension spectra of lines in Euclidean spaces. We highlight three specific directions for further progress. First, it is natural to ask whether the condition on (*a*, *b*) may be dropped from the statement our main theorem: *Does Theorem* 8*hold for arbitrary*\(a,b\in \mathbb {R}\)?

Second, the dimension spectrum of a line \(L_{a,b}\subseteq \mathbb {R}^2\) may *properly* contain the unit interval described in our main theorem, even when \(\dim (a,b)={{\mathrm{Dim}}}(a,b)\). If \(a\in \mathbb {R}\) is random and \(b=0\), for example, then \({{\mathrm{sp}}}(L_{a,b})=\{0\}\cup [1,2]\). It is less clear whether this set of “exceptional values” in \({{\mathrm{sp}}}(L_{a,b})\) might itself contain an interval, or even be infinite. *How large (in the sense of cardinality, dimension, or measure) may*\({{\mathrm{sp}}}(L_{a,b})\cap \big [0,\min \{1,\dim (a,b)\}\big )\)*be?*

Finally, any non-trivial statement about the dimension spectra of lines in higher-dimensional Euclidean spaces would be very interesting. Indeed, an *n*-dimensional version of Theorem 5 (i.e., one in which \(a,b\in \mathbb {R}^{n-1}\), for all \(n\ge 2\)) would, via the point-to-set principle for Hausdorff dimension [8], affirm the famous Kakeya conjecture and is therefore likely difficult. The additional hypothesis of Theorem 8 might make it more conducive to such an extension.

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