Dimension Spectra of Lines

0.21 MB·

by  Neil Lutz

Dimension Spectra of Lines
7 Neil Lutz∗
1
Department of Computer Science, Rutgers University
0
2 Piscataway, NJ 08854, USA
n [email protected]
a
J D. M. Stull†

5 Department of Computer Science, Iowa State University
1
Ames, IA 50011, USA

] [email protected]
C
C January 17, 2017
.
s
c
[

Abstract
1
v
This paper investigates the algorithmic dimension spectra of lines in
8
theEuclidean plane. GivenanylineLwithslopeaandverticalintercept
0
b,thedimensionspectrumsp(L)isthesetofalleffectiveHausdorffdimen-
1
sions of individualpointson L. Wedraw on Kolmogorov complexity and
4
0 geometrical arguments to show that if the effective Hausdorff dimension
. dim(a,b)isequaltotheeffectivepackingdimensionDim(a,b),thensp(L)
1
contains a unit interval. We also show that, if the dimension dim(a,b)
0
is at least one, then sp(L) is infinite. Together with previous work, this
7
implies that the dimension spectrum of any line is infinite.
1
:
v
i 1 Introduction
X
r
a Algorithmicdimensionsrefinenotionsofalgorithmicrandomnesstoquantifythe
densityofalgorithmicinformationofindividualpointsincontinuousspaces. The
most well-studied algorithmic dimensions for a point x ∈ Rn are the effective
Hausdorff dimension, dim(x), and its dual, the effective packing dimension,
Dim(x) [8, 1]. These dimensions are both algorithmically and geometrically
meaningful [4]. In particular, the quantities sup dim(x) and sup Dim(x)
x∈E x∈E
are closely related to classical Hausdorff and packing dimensions of a set E ⊆
Rn [6, 9], and this relationship has been used to prove nontrivial results in
classical fractal geometry using algorithmic information theory [13, 9, 11].
∗ResearchsupportedinpartbyNationalScienceFoundation Grant1445755.
†ResearchsupportedinpartbyNationalScienceFoundationGrants1247051and1545028.
1
Given the pointwise nature of effective Hausdorff dimension, it is natural
to investigate not only the supremum sup dim(x) but the entire (effective
x∈E
Hausdorff) dimension spectrum of a set E ⊆Rn, i.e., the set
sp(E)={dim(x):x∈E}.
The dimension spectra of several classes of sets have been previously investi-
gated. Gu,etal. studiedthedimensionspectraofrandomlyselectedsubfractals
of self-similar fractals [5]. Dougherty, et al. focused on the dimension spectra
of randomtranslations of Cantor sets [3]. In the contextof symbolic dynamics,
Westrick has studied the dimension spectra of subshifts [15].
This work concerns the dimension spectra of lines in the Euclidean plane
R2. GivenalineL withslopeaandverticalinterceptb,weaskwhatsp(L )
a,b a,b
might be. It was shown by Turetsky that, for every n≥2, the set of all points
inRn witheffectiveHausdorff1isconnected,guaranteeingthat1∈sp(L ). In
a,b
recentwork[11], weshowedthatthe dimensionspectrumofa line inR2 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
min{1,dim(a,b)}+1∈sp(L ).
a,b
Togetherwiththefactthatdim(a,b)=dim(a,a2+b)∈sp(L )andTuretsky’s
a,b
result,thisimpliesthatthedimensionspectrumofL containsbothendpoints
a,b
of the unit interval [min{1,dim(a,b)},min{1,dim(a,b)}+1].
Here we build onthat workwith twomain theoremsonthe dimensionspec-
trum of a line. Our first theorem gives conditions under which the entire unit
interval must be contained in the spectrum. We refine the techniques of [11] to
show in our main theorem (Theorem 8) that, whenever dim(a,b) = Dim(a,b),
we have
[min{1,dim(a,b)},min{1,dim(a,b)}+1]⊆sp(L ).
a,b
Givenany value s∈[0,1],we construct,by padding a randombinary sequence,
a value x ∈ R such that dim(x,ax+b) = s+min{dim(a,b),1}. Our second
main theorem shows that the dimension spectrum sp(L ) is infinite for every
a,b
line such that dim(a,b) is at least one. Together with Theorem 5, this shows
that the dimension spectrum of any line has infinite cardinality.
We beginby reviewingdefinitions andpropertiesofalgorithmicinformation
inEuclideanspacesinSection2. InSection3,wesketchourtechnicalapproach
andstate ourmaintechnicallemmas;their proofsaredeferredto the appendix.
InSection4weproveourfirstmaintheoremandstateoursecondmaintheorem,
whose proof is deferred to the appendix. We conclude in Section 5 with a brief
discussion of future directions.
2
2 Preliminaries
2.1 Kolmogorov Complexity in Discrete Domains
The conditional Kolmogorov complexity of binary string σ ∈ {0,1}∗ given a
binary string τ ∈ {0,1}∗ is the length of the shortest program π that will
output σ given τ as input. Formally, it is
K(σ|τ)= min {ℓ(π):U(π,τ)=σ} ,
π∈{0,1}∗
where U is a fixed universal prefix-free Turing machine and ℓ(π) is the length
ofπ. Any π thatachievesthis minimum is saidto testify to, orbe awitness to,
the value K(σ|τ). The Kolmogorov complexity of a binary string σ is K(σ) =
K(σ|λ), where λ is the empty string. These definitions extends naturally to
other finite data objects, e.g., vectors in Qn, via standard binary encodings;
see [7] for details.
2.2 Kolmogorov Complexity in Euclidean Spaces
The above definitions can also be extended to Euclidean spaces, as we now
describe. The Kolmogorov complexity of a point x ∈ Rm at precision r ∈ N is
thelengthoftheshortestprogramπthatoutputsaprecision-rrationalestimate
for x. Formally, it is
Kr(x)=min{K(p) : p∈B2−r(x)∩Qm} ,
where B (x) denotes the open ball of radius ε centered on x. The conditional
ε
Kolmogorov complexity of x at precision r given y ∈Rn at precision s∈Rn is
Kr,s(x|y)=max min{Kr(p|q) : p∈B2−r(x)∩Qm} : q ∈B2−s(y)∩Qn .
When the precision(cid:8)s r and s are equal, we abbreviate K (x|y) by K (x(cid:9)|y).
r,r r
As the following lemma shows, these quantities obey a chain rule and are only
linearly sensitive to their precision parameters.
Lemma 1 (J. Lutz and N. Lutz [9], N. Lutz and Stull [11]). Let x ∈ Rm and
y ∈Rn. For all r,s∈N with r ≥s,
1. K (x,y)=K (x|y)+K (y)+O(logr).
r r r
2. K (x)=K (x|x)+K (x)+O(logr).
r r,s s
2.3 Effective Hausdorff and Packing Dimensions
J. Lutz initiated the study of algorithmic dimensions by effectivizing Hausdorff
dimension using betting strategies called gales, which generalize martingales.
Subsequently, Athreya, et al., defined effective packing dimension, also using
gales[1]. Mayordomoshowedthateffective Hausdorffdimensioncanbe charac-
terizedusing Kolmogorovcomplexity [12], and MayordomoandJ.Lutz showed
3
that effective packing dimension can also be characterized in this way [10]. In
thispaper,weusethesecharacterizationsasdefinitions. Theeffective Hausdorff
dimension and effective packing dimension of a point x∈Rn are
K (x) K (x)
r r
dim(x)=liminf and Dim(x)=limsup .
r→∞ r r→∞ r
Intuitively, these dimensions measure the density of algorithmic information in
the point x. Guided by the information-theoretic nature of these characteriza-
tions,J.LutzandN.Lutz[9]definedthelower andupperconditional dimension
of x∈Rm given y ∈Rn as
K (x|y) K (x|y)
r r
dim(x|y)=liminf and Dim(x|y)=limsup .
r→∞ r r→∞ r
2.4 Relative Complexity and Dimensions
Bylettingthe underlyingfixedprefix-freeTuringmachineU beauniversalora-
cle machine,wemayrelativize the definition inthis sectiontoanarbitraryora-
cle set A⊆N. The definitions of KA(σ|τ), KA(σ), KA(x), KA(x|y), dimA(x),
r r
DimA(x) dimA(x|y), andDimA(x|y)arethenallidenticaltotheir unrelativized
versions, except that U is given oracle access to A.
We will frequently consider the complexity of a point x ∈ Rn relative to a
point y ∈Rm,i.e.,relativetoasetA thatencodesthe binaryexpansionofy is
y
a standardway. We then write Ky(x) for KAy(x). J.Lutz and N. Lutz showed
r r
that Ky(x)≤K (x|y)+K(t)+O(1) [9].
r r,t
3 Background and Approach
Inthissectionwedescribethebasicideasbehindourinvestigationofdimension
spectraoflines. We brieflydiscusssomeofourearlierworkonthissubject,and
we present two technical lemmas needed for the proof our main theorems.
The dimension of a point on a line in R2 has the following trivial bound.
Observation 2. For all a,b,x∈R, dim(x,ax+b)≤dim(x,a,b).
Inthiswork,ourgoalistofindvaluesofxforwhichtheapproximateconverse
dim(x,ax+b)≥dima,b(x)+dim(a,b) (1)
holds. There exist oracles, at least, relative to which (1) does not always hold.
This follows from the point-to-set principle of J. Lutz and N. Lutz [9] and the
existence of Furstenberg sets with parameter α and Hausdorff dimension less
than 1 + α (attributed by Wolff [16] to Furstenberg and Katznelson “in all
probability”). The argumentis simple and verysimilar to our proof in [11] ofa
lower bound on the dimension of generalized Furstenberg sets.
Specifically, for every s ∈ [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).
4
Observation 3. Suppose ax+b=ux+v and u6=a. Then
b−v
dim(u,v)≥dima,b(u,v)≥dima,b =dima,b(x).
u−a
(cid:18) (cid:19)
This observation suggests an approach, whenever dima,b(x) > dim(a,b),
for showing that dim(x,ax + b) ≥ dim(x,a,b). Since (a,b) is, in this case,
the unique low-dimensional pair such that (x,ax+b) lies on L , one might
a,b
na¨ıvelyhopetousethisfacttoderiveanestimateof(x,a,b)fromanestimateof
(x,ax+b). Unfortunately,thedimensionofapointisnotevensemicomputable,
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 [11]). Let a,b,x ∈ R. For all (u,v) ∈ R2 such
that ux+v =ax+b and t=−logk(a,b)−(u,v)k∈(0,r],
K (u,v)≥K (a,b)+Ka,b(x)−O(logr).
r t r−t
Roughly, if dim(a,b) < dima,b(x), then Lemma 4 tells us that K (u,v) >
r
K (a,b)unless(u,v)isverycloseto(a,b). AsK (u,v)isuppersemicomputable,
r r
thisisalgorithmicallyuseful: Wecanenumerateallpairs(u,v)whoseprecision-
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 [11], we used an argument of this type to prove a
general lower bound on the dimension of points on lines in R2:
Theorem 5 (N. Lutz and Stull [11]). For all a,b,x∈R,
dim(x,ax+b)≥dima,b(x)+min{dim(a,b), dima,b(x)}.
The strategy in that work is to use oracles to artificially lower K (a,b)
r
when necessary, to essentially force dim(a,b) < dima,b(x). This enables the
above argumentstructure to be used, but loweringthe 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) > dima,b(x) requires refining the
techniques of [11]. 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).
5
Lemma 6. Let a,b,x ∈ R, k ∈ N, and r = 1. Suppose that r ,...,r ∈ N,
0 1 k
δ ∈R , and ε,η ∈Q satisfy the following conditions for every 1≤i≤k.
+ +
1. r ≥log(2|a|+|x|+6)+r .
i i−1
2. K (a,b)≤(η+ε)r .
ri i
3. For every (u,v) ∈ R2 such that t = −logk(a,b)−(u,v)k ∈ (r ,r ] and
i−1 i
ux+v =ax+b, K (u,v)≥(η−ε)r +δ·(r −t).
ri i i
Then for every oracle set A⊆N,
4ε
KA(a,b,x|x,ax+b)≤2k K(ε)+K(η)+ r +O(logr ) .
rk δ k k
(cid:18) (cid:19)
Lemma 7 strengthens the oracle construction of [11], allowing us to control
complexity at multiple levels of precision.
Lemma 7. Let z ∈Rn, η ∈Q∩[0,dim(z)], and k ∈N. For all r ,...,r ∈N,
1 k
there is an oracle D =D(r ,...,r ,z,η) such that
1 k
1. For every t≤r , KD(z)=min{ηr ,K (z)}+O(logr )
1 t 1 t k
2. For every 1≤i≤k,
i
KD(z)=ηr + min{η(r −r ),K (z|z)}+O(logr ).
ri 1 j j−1 rj,rj−1 k
j=2
X
3. For every t∈N and x∈R, Kz,D(x)=Kz(x)+O(logr ).
t t k
4 Main Theorems
We are now prepared to prove our two main theorems. We first show that,
for lines L such that dim(a,b)=Dim(a,b), the dimension spectrum sp(L )
a,b a,b
contains the unit interval.
Theorem8. Leta,b∈Rsatisfydim(a,b)=Dim(a,b). Thenforeverys∈[0,1]
there is a point x∈R such that dim(x,ax+b)=s+min{dim(a,b),1}.
Proof. Every line contains a point of effective Hausdorff dimension 1 [14], and
by the preservationof effective dimensions under computable bi-Lipschitz func-
tions, dim(a,a2+b)=dim(a,b), so the theorem holds for s=0.
Now let s ∈ (0,1] and d = dim(a,b) = Dim(a,b). Let y ∈ R be random
relative to (a,b). That is, there is some constant c∈N such that for all r ∈N,
Kra,b(y) ≥ r −c. Define sequence of natural numbers {hj}j∈N inductively as
follows. Define h =1. For every j >0, define
0
1
h =min h≥2hj−1 :K (a,b)≤ d+ h .
j h
j
(cid:26) (cid:18) (cid:19) (cid:27)
6
Note that h always exists. For every r ∈N, let
j
0 if r ∈(s,1] for some j ∈N
x[r]= hj
(y[r] otherwise
Definex∈Rtobetherealnumberwiththisbinaryexpansion. ThenK (x)=
shj
sh +O(logsh ).
j j
We first show that dim(x,ax+b)≤s+min{d,1}. For every j ∈N,
K (x,ax+b)=K (x)+K (ax+b|x)+O(logh )
hj hj hj j
=K (x)+K (ax+b|x)+O(logh )
shj hj j
=K (y)+K (ax+b|x)+O(logh )
shj hj j
≤sh +min{d,1}·h +o(h ).
j j j
Therefore,
K (x,ax+b)
r
dim(x,ax+b)=liminf
r→∞ r
K (x,ax+b)
≤liminf hj
j→∞ hj
sh +min{d,1}h +o(h )
j j j
≤liminf
j→∞ hj
=s+min{d,1}.
If 1≥s≥d, then by Theorem 5 we also have
dim(x,ax+b)≥dim(x|a,b)+dim(a,b)
=dim(x)+d
K (x)
r
=liminf +d
r→∞ r
K (x)
=liminf hj +d
j→∞ hj
=s+min{d,1}.
Hence, we may assume that s<d.
LetH =Q∩(s,min{d,1}). We nowshowthatfor everyη ∈H andε∈Q ,
+
dim(x,ax+b)≥s+η−αε, where α is some constant independent of η and ε.
Let η ∈ H, δ = 1−η > 0, and ε ∈ Q . Let j ∈ N and m = s−1. We first
+ η−1
show that
ε
K (x,ax+b)≥K (x)+ηr−c r−o(r), (2)
r r
δ
for every r ∈(sh ,mh ]. Let r ∈ (sh ,mh ]. Set k = r , and define r = ish
j j j j shj i j
forall1≤i≤k. Notethatk isboundedbyaconstantdependingonlyonsand
7
η. Thereforeo(r ) issublinearfor allr . LetD =D(r ,...,r ,(a,b),η)be the
k i r 1 k
oracle defined in Lemma 7. We first note that, since dim(a,b)=Dim(a,b),
K (a,b|a,b)=K (a,b)−K (a,b)−O(logr )
ri,ri−1 ri ri−1 i
=dim(a,b)r −o(r )−dim(a,b)r −o(r )−O(logr )
i i i−1 i−1 i
=dim(a,b)(r −r )−o(r )
i i−1 i
≥η(r −r )−o(r ).
i i−1 i
Hence, by property 2 of Lemma 7, for every 1≤i≤k,
|KDr(a,b)−ηr |≤o(r ). (3)
ri i k
We now show that the conditions of Lemma 6 are satisfied. By inequality (3),
for every 1≤i≤k,
KDr(a,b)≤ηr +o(r ),
ri i k
and so KrDir(a,b) ≤ (η + ε)ri, for sufficiently large j. Hence, condition 2 of
Lemma 6 is satisfied.
To see that condition 3 is satisfied for i = 1, let (u,v) ∈ B (a,b) such that
1
ux+v =ax+b and t=−logk(a,b)−(u,v)k≤r . Then, by Lemmas 4 and 7,
1
and our construction of x,
KDr(u,v)≥KDr(a,b)+KDr (x|a,b)−O(logr )
r1 t r1−t,r1 1
≥min{ηr ,K (a,b)}+K (x)−o(r )
1 t r1−t k
≥min{ηr ,dt−o(t)}+(η+δ)(r −t)−o(r )
1 1 k
≥min{ηr ,ηt−o(t)}+(η+δ)(r −t)−o(r )
1 1 k
≥ηt−o(t)+(η+δ)(r −t)−o(r ).
1 k
We conclude that KrD1r(u,v)≥(η−ε)r1+δ(r1−t), for all sufficiently large j.
Toseethatthatcondition3issatisfiedfor1<i≤k,let(u,v)∈B2−ri−1(a,b)
such that ux+v = ax+b and t = −logk(a,b)−(u,v)k ≤ r . Since (u,v) ∈
i
B2−ri−1(a,b),
r −t≤r −r =ish −(i−1)sh ≤sh +1≤r +1.
i i i−1 j j j 1
Therefore, by Lemma 4, inequality (3), and our construction of x,
KDr(u,v)≥KDr(a,b)+KDr (x|a,b)−O(logr )
ri t ri−t,ri i
≥min{ηr ,K (a,b)}+K (x)−o(r )
i t ri−t i
≥min{ηr ,dt−o(t)}+(η+δ)(r −t)−o(r )
i i i
≥min{ηr ,ηt−o(t)}+(η+δ)(r −t)−o(r )
i i i
≥ηt−o(t)+(η+δ)(r −t)−o(r ),
i i
8
We conclude that KrDir(u,v) ≥ (η−ε)ri+δ(ri−t), for all sufficiently large j.
Hence the conditions of Lemma 6 are satisfied, and we have
K (x,ax+b)≥ KDr(x,ax+b)−O(1)
r r
4ε
≥ KDr(a,b,x)−2k K(ε)+K(η)+ r+O(logr)
r δ
(cid:18) (cid:19)
= KDr(a,b)+KDr(x|a,b)
r r
4ε
−2k K(ε)+K(η)+ r+O(logr)
δ
(cid:18) (cid:19)
4ε
≥ sr+ηr−2k K(ε)+K(η)+ r+O(logr) .
δ
(cid:18) (cid:19)
Thus, for every r∈(sh ,mh ],
j j
αε
K (x,ax+b)≥sr+ηr− r−o(r),
r
δ
where α is a fixed constant, not depending on η and ε.
To complete the proof, we show that (2) holds for every r ∈ [mh ,sh ).
j j+1
By Lemma 1 and our construction of x,
K (x)=K (x|x)+K (x)+o(r)
r r,hj hj
=r−h +sh +o(r)
j j
≥ηr+o(r).
The proof of Theorem 5 gives K (x,ax+b) ≥K (x)+dim(x)r−o(r), and so
r r
K (x,ax+b)≥r(s+η).
r
Therefore, equation (2) holds for every r ∈ [sh ,sh ), for all sufficiently
j j+1
large j. Hence,
K (x,ax+b)
r
dim(x,ax+b)=liminf
r→∞ r
K (x)+ηr− αεr−o(r)
≥liminf r δ
r→∞ r
K (x) αε
r
≥liminf +η−
r→∞ r δ
αε
=s+η− .
δ
Since η and ε were chosen arbitrarily,the conclusion follows.
Theorem 9. Let a,b ∈ R such that dim(a,b) ≥ 1. Then for every s ∈ [1,1]
2
there is a point x∈R such that dim(x,ax+b)∈ 3 +s− 1 ,s+1 .
2 2s
Corollary 10. Let L be any line in R2. T(cid:2)hen the dimensio(cid:3)n spectrum
a,b
sp(L ) is infinite.
a,b
9
Proof. Let (a,b) ∈ R2. If dim(a,b) < 1, then by Theorem 5 and Observa-
tion 2, the spectrum sp(L ) contains the interval [dim(a,b),1]. Assume that
a,b
dim(a,b) ≥ 1. By Theorem 9, for every s ∈ [1,1], there is a point x such
2
that dim(x,ax+b) ∈ [3 +s− 1 ,s+1]. Since these intervals are disjoint for
2 2s
s = 2n−1, the dimension spectrum sp(L ) is infinite.
n 2n a,b
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∈R?
Second,thedimensionspectrumofalineL ⊆R2mayproperly containthe
a,b
unit interval described in our main theorem, even when dim(a,b) = Dim(a,b).
Ifa∈R israndomandb=0,forexample,thensp(L )={0}∪[1,2]. Itis less
a,b
clearwhether thissetof“exceptionalvalues”insp(L )mightitselfcontainan
a,b
interval, or even be infinite. How large (in the sense of cardinality, dimension,
or measure) may sp(L )∩ 0,min{1,dim(a,b)} be?
a,b
Finally, any non-trivial statement about the dimension spectra of lines in
(cid:2) (cid:1)
higher-dimensional Euclidean spaces would be very interesting. Indeed, an n-
dimensional version of Theorem 5 (i.e., one in which a,b∈Rn−1, for all n≥2)
would, via the point-to-set principle for Hausdorff dimension [9], affirm the
famous Kakeya conjecture and is therefore likely difficult. The additional hy-
pothesis of Theorem 8 might make it more conducive to such an extension.
References
[1] Krishna B. Athreya, John M. Hitchcock, Jack H. Lutz, and Elvira Mayor-
domo. Effective strong dimension in algorithmic information and compu-
tational complexity. SIAM J. Comput., 37(3):671–705,2007.
[2] Adam Case and Jack H. Lutz. Mutual dimension. ACM Transactions on
Computation Theory, 7(3):12, 2015.
[3] Randall Dougherty, Jack Lutz, R Daniel Mauldin, and Jason Teutsch.
TranslatingtheCantorsetbyarandomreal. Transactions of theAmerican
Mathematical Society, 366(6):3027–3041,2014.
[4] Rod Downey and Denis Hirschfeldt. Algorithmic Randomness and Com-
plexity. Springer-Verlag,2010.
[5] Xiaoyang Gu, Jack H. Lutz, Elvira Mayordomo, and Philippe Moser. Di-
mension spectra of random subfractals of self-similar fractals. Ann. Pure
Appl. Logic, 165(11):1707–1726,2014.
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