Regularity dimensions: quantifying doubling and uniform perfectness
Abstract.
We study the upper and lower regularity dimensions in relation to the notions of doubling and uniformly perfect. These two regularity properties are closely related which is quantified thanks to the regularity dimensions. The regularity dimensions of pushforward measures onto graphs of Brownian motion are calculated, similarly for pushforwards with respect to quasisymmetric homeomorphisms. We finish by introducing an application to Diophantine approximation in the setting of Kleinian groups.
Key words and phrases:
upper regularity dimension, lower regularity dimension, Assouad dimension, lower dimension, doubling measure, uniformly perfect measure, quasisymmetric homeomorphisms, Brownian motion, Levy process.2010 Mathematics Subject Classification:
primary: 28A80; secondary: 37C45, 28C15, 60G18, 60G51, 60G52, 60J65.1. Introduction
When studying metric spaces and measures on them some basic regularity properties are often assumed. This allows us to avoid many of the common pathological counterexamples. Over time, more sophisticated characteristics have been introduced and studied. We will be interested in studying two regularity properties of measures, doubling and uniform perfectness, and how they interact with concepts of dimension, analogous purely metric regularity properties and each other.
Both spaces and measures can be doubling or uniformly perfect. A metric space is doubling if there is a constant such that can be covered by at most balls of radius for all and . Unless stated otherwise is the open ball of centre and radius . Such spaces are particularly well behaved, for instance Assouad’s embedding theorem says doubling spaces can be embedded into Euclidean space in a nearly biLipschitz way, see [A]. The Assouad dimension can be thought of quantifying how doubling a space is, since a space is doubling if and only if it has finite Assouad dimension. This notion of dimension has also seen much dimension theoretic interest, interacting in a number of ways with the some of the more classical notions of dimension. For an introduction to the Assouad dimension, a formal definition and some recent results see [Fr, FHKY, O, R]. Of particular interest will be the calculation of the Assouad dimension of graphs of Brownian motion in [HY].
In a similar way, a measure on a metric space is said to be doubling if there is a constant such that for all and . Note that if a space is doubling, can be replaced by any in this definition, obtaining new doubling constants . That is
for , and . Here and throughout this paper all measures studied are assumed to be locally finite Borel measures. A natural analogue of the Assouad dimension for measures comes from [KV, LS] and was first studied in [KLV] under the name of upper regularity dimension. The upper regularity dimension of is defined by
taking . Like the Assouad dimension, the upper regularity dimension quantifies how doubling a measure is since a measure has finite upper regularity dimension if and only if it is doubling. This notion of dimension has mostly been studied in the traditional fractal geometry setting, as in [FH, HT, KL]. We will expand on some of the basic properties of this dimension.
Many of the same ideas apply to the concepts of uniformly perfect spaces and measures. A space is uniformly perfect if there exists a constant such that for any there exists . Uniformly perfect sets have been studied for some time, especially in the setting of fractal geometry, [H] contains many interesting references. One can in fact show that a set is uniformly perfect if and only if it has positive lower dimension, the lower dimension being a natural analogue of the Assouad dimension. Many of the previous references for the Assouad dimension discuss the lower dimension if further details are desired.
Being consistent with the previous statements, one can state a definition for a measure to be uniformly perfect; such measures were also recently called reversedoubling in [KL] due to the definition resembling that of doubling measures. Given a measure on a space , is uniformly perfect if there exists a constant of uniform perfectness such that
for all and . As before we can replace by in this definition.
Similarly the lower regularity dimension of a measure is defined by
The lower regularity dimension interacts with the notion of uniform perfectness in much the same way that the upper regularity dimension and the doubling property work together. That is, a doubling measure has positive lower regularity dimension if and only if it is uniformly perfect. This notion can be traced back to [BG] who showed a result similar to that of [KV, LS]. We will explore some properties of the lower regularity dimension and study the links with other notions of regularity, including the upper regularity dimension. Many of the previous references discussing the upper regularity dimension also consider the lower regularity dimension.
Whilst we will not need geometric definitions of the Assouad and lower dimensions, for clarity, one could think of them in terms of the regularity dimensions in the following way, for some bounded metric space ,
2. Results
2.1. Quantifying an example of Heinonen
When studying doubling measures a technical proposition is often employed to truly benefit from the regularity of these measures. Simply put, a doubling measure on a uniformly perfect space is also a uniformly perfect measure. This implies the following important bounds. Say is a doubling measure on a uniformly perfecct, bounded space , then there exists constants and such that for any and
It is not clear where this was first stated, but the standard reference [H] provides this result as an example ([H, Exercise 13.1]) without a proof.
We wish to quantitatively improve this result using the regularity dimensions. More precisely, given a measure of fixed upper regularity dimension on a space of fixed lower dimension , can we bound the lower regularity dimension of as a function of and ? The following result does not quite answer this question as it returns a function of the doubling and uniform perfectness constants. However, as we will see afterwards, this is closer to the desired solution than it appears.
Proposition 2.1.
If is uniformly perfect and is a doubling, fully supported meaasure on then is uniformly perfect. In particular if is uniformly perfect and doubling with doubling constants then
It would have been preferable to obtain a lower bound depending on the lower dimension of and the upper regularity dimension of . Clearly the lower regularity dimension of must depend on the lower dimension of , since the lower dimension is an upper bound, see [BG]. However, as we did not obtain a result depending on the upper regularity dimension, we are lead to ask if there exists a uniformly perfect space and a sequence of doubling measures on that space which all have the same upper regularity dimension but whose lower regularity dimensions can be made as small as possible. A cursory check of some standard examples such as selfsimilar sets and measures implies this might not be feasible.
The question then becomes whether the upper regularity dimension is even distinct from the doubling constants. In [FH], the upper regularity dimension was shown to be bounded above by a function of the doubling constants. The following shows that this bound is actually an equality and thus the upper regularity dimension depends explicitly on the constants. We also obtain an analogous result for the lower regularity dimension.
Theorem 2.2.
Let be a doubling measure fully supported on a metric space with doubling constants , then
Similarly if is a uniformly perfect measure with constants of uniform perfectness then
Combining this result with the previous proposition, it follows that a doubling measure on a uniformly perfect space must have lower regularity dimension bounded below by a function of the upper regularity dimension and the constant of uniform perfectness. However, as the above is concerned only with the infimum over all and the formula in Proposition 2.1 relies on a specific , we cannot find an exact formula linking the lower regularity dimension and the upper regularity dimension in our setting.
We finish this section with a brief discussion of the sharpness of this result. Assuming positive lower dimension of the space is required here as it is simple to construct spaces of zero lower dimension but finite Assouad dimension. In such a setting there must exist a measure of upper regularity dimension close to the Assouad dimension of the space and so doubling. But any measure on this space can not be uniformly perfect as the lower regularity dimension is a lower bound to the lower dimension. A trivial such example would be the set of points with the Euclidean metric. This set is know to have zero lower dimension but full Assouad dimension; doubling measures on this space were explicitly constructed in [FH].
There are many examples of uniformly perfect measures, even on doubling spaces, that are not doubling so we cannot interchange the two notions and obtain an analogous result. For instance, one can take a selfsimilar set with overlaps, this is a doubling space. There are numerous uniformly perfect measures on such a space that are not doubling, as can be seen in [HT]. Thus a uniformly perfect measure on a doubling and uniformly perfect space need not be doubling.
2.2. Regularity dimensions under quasisymmetric homeomorphisms
Quasisymmetric homeomorphisms are a generalisation of biLipschitz maps, preserving relative sizes but not necessarily global size which were first introduced in [AB, TV]. In the Euclidean setting quasisymmetric homeomorphisms are equivalent to the often studied quasiconformal homeomorphisms. In this article the metric of a given metric space is denoted . A homeomorphism is an quasisymmetric homeomorphism if there is a homeomorphism such that
implies
for all and for all .
Equivalently there exists a homeomorphism as above such that
for any distinct points . Here is not unique for a given quasisymmetric homeopmorphism.
A property of particular interest to us is that doubling and uniform perfectness of spaces are quasisymmetric invariants. This can be quantified, so there are bounds on the Assouad and lower dimensions of images of spaces under quasisymmetric embeddings, see [H] for further details. We wish to know if the same holds for doubling and uniformly perfect measures. In particular we will study pushforward measures under quasisymmetric homeomorphisms. Given a measure on a space and a map from to some space , the pushforward measure of under is denoted and is defined by
for any measureable subset of , where .
To avoid having trivial upper and lower regularity dimensions of it is reasonable to assume that is doubling and uniformly perfect. This then lets us employ the following theorem.
Theorem 2.3 ([H, Theorem 13.11]).
A quasisymmetric homeomorphism of a uniformly perfect space is quasisymmetric with of the form
where and depend only on and .
For clarity we will often write to indicate the homeomorphism associated with the constant as described here. Section 3 of [TV] proves this result and explicitly calculates .
Theorem 2.4.
Let be a uniformly perfect space and be doubling on . When is an quasisymmetric homeomorphism the following bounds hold
and
where is the pushforward of .
2.3. Pushforwards of measures onto graphs of Brownian motion
Having shown that the regularity dimensions are well behaved under quasisymmetric maps we now turn our attention to random maps and ask if doubling and uniform perfectness are also preserved in these situations. Specifically we will consider maps from the unit interval onto graphs of Lévy processes.
A Lévy process is a random function satisfying:

with probability 1, ;

is right continuous and has left limits at every point ;

is equal to in distribution for all (stationary increments);

for all , the increments are independent;
When the distribution of increments is chosen to be the Normal distribution with mean 0 and variance we recover the Wiener process (or Brownian motion). Lévy processes are fundamental tools in several areas of mathematics and have been extensively studied. They were first introduced by Lévy in [L]. Their fractal properties were first investigated by [T] where the Hausdorff dimension of the graph of Brownian motion was shown to be almost surely equal to and the range of dimensional Brownian motion was found to have dimension almost surely for any .
Whilst many geometric objects associated to Lévy processes have been studied, we will only consider properties of the graphs of Lévy processes. Given a Lévy process , the graph of restricted to the unit interval is defined by:
More generally, denote by the graph of the process restricted to the interval . There is a naturally associated map which maps the unit interval to the graph of the process, that is . This will be the map we wish to use to construct pushforward measures, and for the rest of this section should be assumed to be this map.
One of the interesting features of Lévy processes is their statistical selfaffinity. Not all processes have this property so we will restrict to stable or stable processes, that is for some
for all and means equal in distribution. For example the Wiener process is 2stable. In fact all stable processes have .
Our final condition is a simple assumption that the distribution is nonvanishing on . Nonzero on an interval would also work, this is just to ensure the graphs are not just multiple flat lines, such as in a Poisson process.
This leads us to the question of this section: given a doubling measure on the unit interval, is also doubling? A similar question holds for uniformly perfect measures. In [HY], it was shown that the Assouad dimension of is almost surely 2 so there must exist at least one doubling measure on the graph. However, most measures on the graph might not even be doubling. For the Hausdorff dimension, the proof by Taylor shows that the Hausdorff dimension of the pushforward of Lebesgue measure almost surely attains the dimension of the graph itself. It turns out that this is usually not the case for the regularity dimensions.
Theorem 2.5.
Let be a doubling measure on and a stable Lévy process with the distribution being non vanishing on . Then is almost surely not doubling on . Also, is almost surely not uniformly perfect.
Trivially this implies the upper regularity dimension of is almost surely infinity and the lower dimension is almost surely zero. Therefore any measure whose upper regularity dimension approximates the dimension of the graph is highly dependent on the specific graph and so there is no one measure that attains the dimension for typical realisations, unlike the Hausdorff case.
2.4. Uniformly perfect and weakly absolutely decaying measures
It is known that the lower regularity dimension of a measure is positive if and only if the measure is uniformly perfect, see [KL]. A property that has appeared recently in Diophantine approximation is the notion of weakly absolutely decaying. This was first introduced in [BGSV] following the previous uses of friendly measures by [KLW] and quasidecaying measures by [DFSU, DFSU2]. A measure is weakly absolutely decaying for some when there exists constants such that for all
for all and .
This property has some resemblance to the ideas of doubling and uniformly perfect measures. We show the following.
Proposition 2.6.
If a measure has positive lower regularity dimension then
This result actually leads to an equivalent but hopefully more applicable statement of Theorem 2 in [BGSV]. Their theorem concerns the size of the set of points which well approximate a given point with respect to some function for limit sets of Kleinian groups . In the Euclidean setting, one usually considers the Lebesgue measure of this set. However when working on different sets other measures must be considered that somewhat mirror the regularity of Lebesgue measure. This leads to the notion of decaying. Without further elaborating on the definitions we state both the old and new versions of the theorem.
Theorem 2.7 ([Bgsv, Theorem 2]).
Let be a nonelementary, geometrically finite Kleinian group and let be a parabolic fixed point of , if there are any, and a hyperbolic fixed point otherwise. Fix , and let be a compact subset of equipped with a weakly absolutely decaying meausre . Then
Theorem 2.8.
Assume and are as above. Let be a compact subset of with lower dimension equal to . For any , if there exists a weakly absolutely decaying measure on , then
In particular, if then any weakly absolutely decaying measure on is such that . One can also find a sequence of weakly absolutely decaying measures on such that .
An advantage of writing the theorem with respect to the lower dimension of the limit set is that the lower dimension of limit sets were calculated in [Fr2]. Therefore, given a limit set, we can quickly check if there will be measures that are weakly absolutely decaying such that the sum in the theorem converges for the given . In [Fr2], Fraser also calculated the regularity dimensions of PattersonSullivan measures, providing us with explicit measures that could be used in the theorem, as the upper and lower regularity dimensions of PattersonSullivan measures are strictly positive and finite.
Theorem 2.8 follows from Proposition 2.6 and Theorem 2.7. As such we do not provide a proof here, for the interested reader the proof of Theorem 2.7 in [BGSV] is very accessible.
Weakly absolutely decaying measures were the correct measures to consider in the setting of limit sets of Kleinian groups whereas friendly measures were used in the context of subsets of Euclidean space. It would be a natural extension to study the links between friendly measures and the regularity dimensions, especially given that one of the conditions for a measure to be friendly is that it is doubling.
3. Proofs
This section will be broken into several subsections that are mostly independent of each other but the notation will remain consistent throughout. In section 3.1 we cover the results found in section 2.1. In section 3.2 we prove Theorem 2.4. Section 3.3 is dedicated to measures on graphs of Lévy processes. Finally in section 3.4 a short proof of Proposition 2.6 is provided.
3.1. Quantifying an example of Heinonen
Proof of Proposition 2.1.
Let and be as in the statement of the proposition with constant of uniform perfectness and doubling constants . We will rework the proof found in [KLV, lemma 3.1], paying careful attention to the constants in play. Note there is another proof in [RS, Lemma 4.5] which could lead to different bounds, but we do not pursue this here.
To start, a technical result is required. Proposition B.4.7, in [G], states that in our setting there exists a constant such that
for any . We start by determining as a function of our known constants.
For any and , as is uniformly perfect, there exists such that
This choice of ensures that . Thus and
Recall is the doubling constant of where . By iterating this construction we obtain
for any , as desired.
Returning to the actual question, fix , and choose such that so that . Then
as desired.
∎
Note that in the proof of [G] one should use the optimal doubling and uniform perfectness constants to obtain the best bound possible, however the result itself is likely not sharp.
Now we turn our attention to the relationship between the doubling constants and the upper regularity dimension, as well as the constants of uniform perfectness and the lower regularity dimension.
Proof of Theorem 2.2.
We start by proving the link between the upper regularity dimension and the doubling constants. The upper bound follows from [FH], the difference in formula is purely notational. To obtain a lower bound on the upper regularity dimension of a measure on a space , it suffices to find, for and , with , such that , a sequence of
From the definition of doubling we know that for all . Fixing we pick to be reasonably sharp in the sense that there exists at least one pair of such that .
Recall and we simply pick any sequence of increasing . Then from our choice of , the pair and are the pair obtained above. is then fixed by . Finally, due to the choice of , . To choose our sequence of
completing the proof.
The lower regularity dimension result follows similarly. ∎
3.2. Regularity dimensions under quasisymmetric homeomorphism
Whilst Theorem 13.11 of [H] is the key ingredient in the proof of our theorem, the following proposition which can also be found in [H] is also required.
Proposition 3.1 ([H, Proposition 10.6]).
When a quasisymmetric homeomorphism is quasisymmetric, its inverse is an quasisymmetric homeomorphism with given by for .
It is then clear that a quasisymmetric homeomorphism on a uniformly perfect space is associated with a homeomorphism and is also a quasisymmetric homeomorphism associated with the function . Note that the homeomorphism is not exactly , but, as it is a upper bound to the desired function, will be an quasisymmetric homeomorphism.
Proof of Theorem 2.4.
We start by proving the upper bound for the upper regularity dimension. Let and . Since is uniformly perfect, we can find such that and . Without loss of generality, choose so that , this will be required to use the exact formula for .
Choose any point . From our choice of , it is clear that . Thus, as is quasisymmetric , and so
Similarly, choosing , we have and so
Hence
Similar statements clearly hold for with . Thus, for any ,
where is the constant from the definition of the upper regularity dimension of with respect to . As is arbitrarily chosen this completes the upper bound.
For the lower bound we can repeat the above argument, swapping with and with . Due to the correspondence between and we see that as desired.
Proofs for the lower regularity dimension follow similarly.
∎
3.3. Pushforward of measures onto graphs of Brownian motion
Choose a Lévy process which satisfies the conditions in Theorem 2.5 with scaling coefficient and fix a graph realised by this process. Start by assuming , the proof will work in the same way for given a slight modification which will be commented on later in the proof. is taken to be a doubling measure on the unit interval. Recall is defined to be the function which maps the unit interval to the graph of our Lévy process and is the pushforward measure of onto the graph that we wish to study.
We start by calculating the almost sure upper regularity dimension of . Let . The general strategy for this proof is to find a sequence of events that are all independent and have positive probability. Then a simple application of the BorelCantelli lemma will yield that almost surely these events will happen infinitely often. By choosing our events carefully this will yield a sequence of balls that show the upper regularity dimension of the pushforward measure must be greater than . As is abitrary, this will conclude the proof.
Given our scaling Lévy process, we define the rectangle centered at with side lengths by where is just an interval of length and centre . will denote the graph of above the interval .
The particular events we are interested in are defined as follows: let , and , then is the event in which and . These events are chosen so that the measure on the graph will be ‘large’ on the rectangle of small side length but ‘small’ on the rectangle of small side length . Figure 4 is a geometric representation of such an event.
Note that here is important for the rectangles to be tall and thin. If it would suffice to change to and similarly for the smaller rectangles. The rest of the proof would run in the same way afterwards with some slight changes in the calculations of at the end.
Given any sequences , and we can consider the associated events as above. To make sure the ‘smaller’ rectangle is actually smaller, assume without loss of generality. If for all , then the events are all independent due to the independent increment property of the Lévy process. As long as the distribution of is nonvanishing on the unit interval, the probability of any of these events is positive.
We can now choose our sequence of events. Start by picking any disjoint and strictly increasing sequence of reals , would suffice. Then the are taken so that the intervals do not overlap ensuring independence, say . Initially any sequence of can be chosen as long as and, again, for each . will be fixed later, for now it is just a real greater than 1. As the process is scaling one can map onto the unit square via an affine map and the image of the graph under this transformation, denoted , will have distribution equal to the original distribution as it is scaled following the definition of scaling, so in distribution. Therefore the probability of an event is equal to the probability the graph of stays in the unit square and
Thus the probability of depends solely on the ratio . If diverges then the conditions for BorelCantelli are satisfied and the argument continues. However, if not, the sequence is modified in the following way. Each gives us a ratio and a probability . Construct a function such that for all . Then, keeping fixed, change the so that each ratio is repeated many times. For instance, if then and are chosen so that and all give the same and then is chosen with respect to etc. The new sequence is constructed such that diverges, satisfying the conditions for BorelCantelli.
Hence, by the BorelCantelli lemma, infinitely many occur with probability one. So there are sequences , and such that, with full probability, all of the events happen and .
Given a specific event we wish to consider the measure of the rectangles. The ratio of measures of such rectangles is determined by the original measure on the interval. We let , this is just to have a number for which the following bound holds but is also fixed and positive due to Proposition 2.1. Thus we obtain the following bound:
where comes from the definition of the lower regularity dimension.
The only variable left to be fixed is . We wish to have the above ratio greater than . After a short calculation, it is clear that this is always true if . Thus by choosing such a we have
To show the upper regularity dimension is greater than we need to consider balls not rectangles. Thankfully due to our construction and . Hence
completing the proof.
For the lower regularity dimension it suffices to change the events in the following way. Assuming , let , and , then is the event where and . The previous argument then works in much the same way, showing that the lower regularity dimension of is zero as desired.
3.4. Uniformly perfect and weakly absolutely decaying measures
Proof of Proposition 2.6.
If is weakly absolutely decaying then for any and . Thus
and so .
For the other direction, assume . Then for any , there exists such that for all and
Given and , choose . Inserting this value of into the above yields
Hence
and so is decaying. ∎
Acknowledgements
The author was financially supported by an EPSRC Doctoral Training Grant (EP/N509759/1). He would like to thank Jonathan Fraser for many helpful conversations.
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