On planar selfsimilar sets with a dense set of rotations
Abstract.
We prove that if is a planar selfsimilar set with similarity dimension whose defining maps generate a dense set of rotations, then the dimensional Hausdorff measure of the orthogonal projection of onto any line is zero. We also prove that the radial projection of centered at any point in the plane also has zero dimensional Hausdorff measure. Then we consider a special subclass of these sets and give an upper bound for the Favard length of where denotes the neighborhood of the set .
2000 Mathematics Subject Classification:
28A801. Introduction
In this paper we investigate the orthogonal and radial projection properties of some selfsimilar sets in the plane. Planar self similar sets are the attractors of iterated function systems whose maps are contracting similitudes. By a classical result of Marstrand [6], if is a planar Borel set with (where denotes the Hausdorff dimension) then in Lebesgue almost all directions the orthogonal projections onto lines have Hausdorff dimension . If then almost all projections have positive Lebesgue measure. Therefore the natural question is to ask when the projections have positive dimensional Hausdorff measure in the case when .
It is useful to partition planar selfsimilar sets sets into two categories when studying their orthogonal projections onto lines, namely, the case when the similitudes do not involve rotations or reflections, and the others. The sets whose defining maps do not involve rotations are distinguished from the others with the relatively simple structure of their projections: These projections are selfsimilar sets in . The defining maps for the projections can be viewed as a family of maps depending on a parameter (the projection angle) and measure theoretic arguments can be made about the properties of “typical” projections. We refer to [10] for the details and some applications of this method.
In the case when rotations are involved, the projections are no longer selfsimilar, thus it is significantly more complicated to study their structure. Our main result is concerned with the case when the defining maps generate a dense set of rotations, and partly answers a question by Mattila in [8]. This would be the case when, for example, one of the maps involved rotation by an irrational multiple of . In fact, if none of the maps involve reflections, our condition is equivalent to having at least one map with an irrational rotation. The idea of the proof is that, if the set of rotations is dense, there are many groups of smaller copies of the original set that are approximately the same size and aligned in a dense set of directions. This is shown by modifying a “doubling” argument that has been used in [9]. These copies “pile up” above a typical point in the projection, making the density of the projected measure infinite. Density of directions suggests that this is the case in a dense set of directions, and hence, by approximation, all directions. We prove the following:
Theorem. Let be a selfsimilar set whose defining maps generate a dense set of rotations modulo . If is the similarity dimension of , then for all lines , the dimensional Hausdorff measure of the orthogonal projection of onto is zero.
The case when the rotations are in a discrete set of directions is still an open question.
Finally, in the last section we give an upper bound for the Favard length of the neighborhood of where is a homogeneous selfsimilar set of similarity dimension whose defining maps include two nonrotating maps and a (nonreflecting) rotation by a Diophantine multiple of . A number is called Diophantine if there exist such that for any two integers and (we say the number is Diophantine in this case). Recall also that a selfsimilar set is called homogeneous if all defining maps have the same contraction factor. The Favard length of a planar set is given by
where is the Lebesgue measure of the orthogonal projection of onto , the line through the origin making angle with the positive axis. Besicovitch’s projection theorems tell us that in the above case, the Favard length of converges to zero since is irregular, but in very few cases we have precise estimates for the decay rate. A set is called a 1set if . There is the known lower bound when is any Borel 1set in the plane (see [7]).
We will prove the following theorem:
Theorem. Let be the attractor of a homogeneous iterated function system with similarity dimension 1 which produces two nonrotating maps of the same contraction factor and a map rotating by angle , where is Diophantine. Then, denoting the orthogonal projection of onto by , for any there exists such that
(1) 
uniformly for all , thus
(This theorem will be stated more precisely in section 3 (see Theorem 3.1).
In [11], it was proven that for selfsimilar sets of similarity dimension with strong separation and without rotation, the bound
(2) 
holds for some , where
(3) 
The class of sets we study in this paper are the only selfsimilar sets that are currently known to obey a bound better than (2).
2. Projections of planar selfsimilar sets with a dense set of rotations
We first introduce the terminology before we state the result:
An iterated function system is a finite collection of Lipschitz maps on with Lipschitz constants less than 1.
Consider an IFS in the plane with maps . There is a unique nonempty compact set , called the attractor of this iterated function system, that satisfies
(4) 
We say is selfsimilar if the maps are similitudes. The set can be viewed as the image of a “projection” from a symbol space , given by
The limit is a singleton since the maps are contractions. Finite sequences of symbols will be called words. We will use to denote the set of sequences in starting with the word (such sets are called cylinder sets).
Now assume is selfsimilar. Let be the line through the origin making an angle of with the positive axis and let be the composition of with the orthogonal projection onto . Assume each has contraction rate . The similarity dimension of the system above is the number such that
We will shortly call the similarity dimension of , when the iterated function system in question is clear from the context (Note that can be produced by different iterated function systems). It is wellknown that . For many purposes the interesting situation is the case when . The BandtGraf condition [2] and the open set condition (see [4]) are examples of necessary and sufficient conditions for this to be true.
For a word define . Define
We denote by the infinite sequence The length of will be denoted by .
Each is either a scaling and reflection composed with a translation, or a scaling and rotation composed with translation. Given a word we can write
where if contains a reflection (about the line ), or if it contains a rotation only (by the angle ). Note that if then , and for any words . In this paper we will consider angles modulo , e.g. we will write for . We are assuming that the set
is dense modulo . Our result is as follows:
Theorem 2.1.
Let be a selfsimilar set such that is dense modulo . If is the similarity dimension of , then for all lines , the dimensional Hausdorff measure of the orthogonal projection of onto is zero.
Note that if is dense, then is also dense. Given an we will denote by some fixed word such that and .
Let be the similarity dimension of . We define to be the product measure on , that is . Let .
Definition 1.
Two words are called relatively close if

;

and ;

There exists such that .
This definition means the smaller copies of the set defined by and have relatively the same size and orientation, and the convex hulls of their projections on have large overlap.
The proof of Theorem 2.1 is based on a sequence of lemmas:
Lemma 2.2.
Given any , an angle and word , for almost all points there exist infinitely many words such that is a prefix of , and .
Proof.
Let . There is such that the set
forms an net modulo . Given any word , if we can concatenate a sequence of ’s of length no more than so that and . If there is an orientation reversing and then we concatenate to a suffix of ’s as above to get and . So we conclude that there is an integer such that given any word we can find a suffix of length no more than satisfying and .
Given this , there exists such that if is a word of length no more than then .
Let be the set of points in where the claim fails. Given , let be the points of for which no prefix of length exists as in the claim. Then . Therefore it suffices to prove that for any .
Now we fix . Write as a disjoint union of cylinders represented by words length bigger than and let be the set of words corresponding to the cylinders in this union. For each word , let be a word of length no more than such that and . Let . Then . Given , define in the same way: Write as a disjoint union of cylinders and let be the set of corresponding words. For each word in , find a word as above. Define . Note that for all . Since is a subset of and , the result follows. ∎
The following corollary will be useful when studying visibility properties:
Corollary 2.3.
Let be any function. Then, given a word and , for almost all there exist prefixes such that is a prefix of , and .
Proof.
Let be a finite collection of angles that form an net modulo . Then, given a fixed , by Lemma 2.2 almost all have prefixes such that is a prefix of , and . Since forms an net, this implies that almost all have prefixes satisfying the conditions of our claim. ∎
A set of real numbers is called arithmetic if all numbers in the set are integer multiples of and is the biggest number with this property. We quote a probabilistic lemma (see [5], Vol II, Lemma V.4.2):
Lemma 2.4 (Feller).
Let be a distribution in concentrated on but not at the origin, and the set formed by the points of increase of . If is not arithmetic, then is asymptotically dense at infinity in the sense that for given and sufficiently large, the interval contains points of . If has arithmetic support then contains all points for sufficiently large.
Lemma 2.5.
Given any and a function , there exist such that are distinct and relatively close with . Moreover, in addition to the conditions (i)(iii) of Definition 1, we can choose and in such a way that: (iv) , , and (v) there exists such that .
Proof.
Note that (v) trivially implies (iii), so it suffices to check condition (v) alone to prove (iii). Let . Given , let be the the set of all cylinders with
Let . We first consider the case when is not arithmetic: Let be a distribution supported on . Find sufficiently large as in Lemma 2.4 for . Let be such that
Then each has a subcylinder (i.e. is a prefix to ) such that . So there exist at least cylinders such that for each pair . Since as , by choosing small enough we can also find a pair such that and . If then let and . If and is an orientation reversing map in the iterated function system, let and . Then and , thus (i) and (ii) in Definition 1 are satisfied.
In the case when is arithmetic for some , we again consider a distribution supported on , and using the lemma find such that for all . Let be given by
Then there exists a subcylinder of with . This is true for each . The rest of the argument is as in the nonarithmetic case; we can find satisfying the first two conditions of Definition 1.
We set . Choosing to be plus the direction of the line segment joining to and setting , (v) is satisfied (if these two points coincide then any direction can be chosen as ). For (iv), consider the pairs
All these pairs satisfy (i)(ii) and (v) (hence (iii)) with the same , and . Since , some of them will also satisfy (iv) (see Fig. 2). ∎
Lemma 2.6.
Given any and , there exist distinct words and such that the are mutually relatively close and .
Proof.
The statement is true for by the previous lemma. We will now prove that if the statement is true for then it is also true for .
So we now assume that the claim of the lemma is true for . Find and for . Let . Choose so small that
(5) 
and
(6) 
Consider the function . Apply Lemma 2.5 with and to find and satisfying conditions (i)(v) of the Lemma, that is,

;

and ;

;

and .
We now claim that the distinct words are mutually relatively close.
Since , it is easily seen that (i) and (ii) of Definition 1 are satisfied with . Clearly, are mutually relatively close. We now prove that they are also relatively close:
Without loss of generality, we can assume . Note . Let be as described in part (iii) of Definition 1. Denote by the unit vector in the plane in the direction . Observe that . Then for any and , using (5) and (d) we get (see Fig. 3)
(7) 
Therefore are mutually relatively close. A similar proof shows that are also mutually relatively close. Let be such that . Now we will prove that and are relatively close and that we can use any in Definition 1 (iii). From this the result will follow, since for we get
(we can use any as if ).
Denoting by the rotation map in the plane by angle , for any word with we can write
for some vector . Also recall that for any by definition. Using (6), the conditions (a)(c) above and the linearity of we get (see Fig. 4)
and the result follows since for any . The lemma is proved.
∎
And finally we prove the following result which immediately implies the main result of this section:
Proposition 2.7.
For all we have .
Proof.
Let be the projection of the measure under . We will prove that given any , at almost all points the dimensional upper density of given by
is at least , where is a constant independent of (here denotes an open ball of radius around ) . The result will follow from a standard density theorem (see [4] Prop. 2.2 for a statement).
Observe that if are distinct and relatively close for some and , then and can not be subwords of each other provided is small enough. This follows from the fact that if any of and is a proper subword of the other. Note that this also implies that and are disjoint in . Now, given , we can find distinct that are mutually relatively close for some . This we can see by applying Lemma 2.6 with any . By using small enough in the Lemma if necessary, we can assume that are disjoint. For the purposes of our proof, the words will be regarded as mutually relatively close.
Now let where satisfies the conditions of Lemma 2.2 with , word and . That is, has prefixes of the form where satisfy . By Lemma 2.2, almost all are of this form. It follows that for each , and , using ideas similar to those in (7) we get
Thus if we set then each lies in , thus (using that for each ):
and this proves our claim. ∎
Applications to visibility. We now make remarks about the application of the method above to visibility problems. We consider the visibility of a selfsimilar set from a point. Given a point , we define the radial projection (centered at ) as
Definition 2.
We say that a set is visible from if . We say is invisible from if .
Remark. The standard terminology is to say “visible/invisible from ” in the case .
We observe that if satisfies the conditions of Theorem 2.1, we can use the radial projections of the measure to show that, at all points, the projected measure has infinite dimensional density almost everywhere (where is the similarity dimension). More precisely, given , we define to be the restriction of to and define as the projection of to a measure on via . It is easy to see that, in the proof of Lemma 2.7, one can use Corollary 2.3 and the same idea of aligning the 1relatively close squares around typical points in such a way that the radial projections of these squares have large overlap, making the measure density big. Since the required modifications are fairly obvious, we state the result without a proof:
Theorem 2.8.
Let be a selfsimilar set for which is dense modulo . If is the similarity dimension of , then for all points in the plane, is invisible from .
We also remark that in a recent work [12] of K. Simon and B. Solomyak, it is proven that purely unrectifiable planar selfsimilar sets with finite measure and satisfying the open set condition are invisible from all points in the plane. This implies our conclusion in the case . To see this, note that if and , dense set of rotations implies the nonexistence of tangent directions at all points of , hence the pure unrectifiability of (see [3] for an overview of tangency properties).
3. Favard length in some special cases
The Favard length of a planar set is defined as
where is the orthogonal projection of onto . It can be interpreted as a measure of the probability of Buffon’s needle hitting the set . By Besicovitch’s wellknown theorem, irregular 1sets project to zero measure in Lebesgue almost all directions, therefore their Favard length is zero (see [3] for details). One can then ask about the speed at which decreases to zero, where denotes the neighborhood of the set .
Mattila has shown [7] that is a lower bound for 1sets. The question about the best possible (general) upper bound is open. We should remark that different upper bounds may apply to different classes of sets.
In [11], Y. Peres and B. Solomyak gave an upper bound for selfsimilar 1sets in the plane that do not involve rotations and satisfy the strong separation condition. Defining as in (3), they proved that for such sets for some is an upper bound. This bound is of course far from the lower bound given by Mattila, and it is not known whether it can be improved or not. Peres and Solomyak also gave an example of a random Cantor set for which the expected upper bound is of the same order as Mattila’s lower bound. It is an interesting problem to give more accurate estimates for deterministic sets. The method used in this section is based on the approach in [11].
Now we return to our second main result. We are going to consider a homogeneous selfsimilar set in the plane of similarity dimension , defined by maps. Let be the common contraction factor of the homogeneous system. We are assuming that the defining maps produce two nonrotating maps of the same contraction rate and a map containing rotation by a Diophantine multiple of . By composing the nonrotating maps with themselves to remove reflections if necessary, we can assume that

There are two distinct words with , and ;

One of the defining maps, say, , contains irrational rotation by , where is Diophantine.
Under these assumptions, we are going to prove the following:
Theorem 3.1.
Given any , there exists such that
(8) 
uniformly for all , where . Thus,
Observe that it suffices to prove this theorem for a sequence of decreasing to such that is (this will only change the constant ). For a word , we have since the system is homogeneous. is bounded. Also for simplicity, we will assume that the diameter of
Before we begin the proof, we mention an equivalent reformulation of this result: Let be the convex closure of the selfsimilar set . Let
Observe that there exists a constant independent of and translation maps in the plane such that, for , can be covered by these translates of . This follows from the fact that contains a (nontrivial) ball since there are irrational rotations. Therefore, is comparable (with uniform constants) to . Note that if we take , we have therefore (8) becomes
(9) 
In this formulation, the lower bound given for by Mattila’s result is .
The first stage in the proof is to construct relatively close words for any given : Let as in condition (a) given above. Let . Note that and satisfy the definition of relative closeness if we choose and to be perpendicular to the line connecting to (or any line if these points coincide). Now we observe that are also mutually relatively close words with the same and used for and : Clearly and . And since contain no rotation, the points and lie on the line connecting to . Continuing this procedure, for any , we can obtain words that are mutually relatively close and with for all .
By our assumption is Diophantine, that, is for any integers we have
(10) 
Observe that, given any , by the pigeonhole principle, there are integers and such that