Invisibility via reflecting coating
Abstract.
We construct a subset of the unit disc with the following properties. (i) The set is the finite union of disjoint line segments. (ii) The shadow of is arbitrarily close to the shadow of the unit disc in “most” directions. (iii) If the line segments are considered to be mirrors reflecting light according to the classical law of specular reflection then most light rays hitting the set emerge on the other side of the disc moving along a parallel line and shifted by an arbitrarily small amount.
We also construct a set which reflects almost all light rays coming from one direction to another direction but its shadow is arbitrarily small in other directions, except for an arbitrarily small family of directions.
1. Introduction and the main result
We will construct two prefractal sets with special reflective and projection properties. First, we will construct a subset of the unit disc with the following properties. (i) The set is the finite union of disjoint line segments. (ii) The shadow of is arbitrarily close to the shadow of the unit disc in most directions (that is, the complement of the set of such directions has arbitrarily small measure). (iii) If the line segments are considered to be mirrors reflecting light according to the classical law of specular reflection then most light rays hitting the set emerge on the other side of the disc traveling along a parallel line and shifted by an arbitrarily small amount.
Next, we will construct a set which reflects almost all light rays coming from one direction to another direction but its shadow is arbitrarily small in other directions, except for an arbitrarily small family of directions.
The article has multiple sources of inspiration. One of them is recent progress on invisibility (see [Uhl09, GKLU09]) although there is no direct relationship between our article and those papers at the technical level. Another source of inspiration is the theory of radiative transfer (see [Per02]) which is used by astrophysicists to study the scattering of light. The relationship between reflections and “visibility” is implicit in Problem 2.6 of [ABS12]. And last but not least, Falconer’s “digital sundial” theorem ([Fal03, Thm. 6.9]; see also Proposition 2.1 below) provided not only inspiration but also a significant mathematical step in our argument.
The set constructed in Theorem 1.3 below might be a building block of a reflective “surface” that is the subject of Problem 2.6 of [ABS12]. At this point, we do not know how to turn this observation into a rigorous solution of that problem. The set in Theorem 1.3 resembles the set in the Besicovitch theorem (see [Fal86a, Thm. 6.15, p. 90]) and the “fourcorner Cantor set” (see [PSS03, BV10, NPV10]) but we did not find a way to turn these classical constructions into a direct proof of our result.
We now introduce notation and definitions needed to state our results in a rigorous way. It will be convenient to identify and and switch between the vector and complex analytic notation. Let denote the orthogonal projection of a set on the line . The 1dimensional Lebesgue measure will be denoted . By abuse of notation, we will use the same symbol to denote 1dimensional measures on lines in and restriction of the Lebesgue measure to . Let denote the unit disc.
We will be concerned with light rays moving at a constant speed and reflecting from “mirrors,” that is, line segments. We will describe trajectories of light rays using “directed lines”. A directed line is an affine map , , where , and . We will write . Let be the collection of all pairs such that , and . The set may be considered to be a subset of . As such, it inherits the usual metric and topology from . The following formula uniquely defines a measure on ,
(1.1)  
Let be the set of all pairs such that . Then . Heuristically speaking, according to the probability measure , the direction of is chosen uniformly, and so is the distance from (within ).
A light ray is a continuous and piecewise affine map . We require that for each light ray, can be partitioned into a finite number of intervals and on each interval, , for some and depending on the interval. Either one or two of the intervals have infinite length. We will say that comes from direction if on the interval that extends to . We will say that escapes in direction if on the interval that extends to . We will assume that light rays reflect from mirrors (line segments) according to the classical law of specular reflection in which the angle of reflection is equal to the angle of incidence. Suppose that a set consists of a finite number of line segments. Then a light ray which arrives “from infinity” and hits will not be trapped (this can be proved as Proposition 2.1 in [ABS12]) so it has a well defined direction from which it arrives and a direction of escape. If a light ray which comes from direction escapes in the direction after reflecting in mirrors comprising then we will write .
The following is the first of our two main results.
Theorem 1.1.
(Invisibility via reflecting coating) For every there exists a set which consists of a finite number of line segments and such that there exists a set with , satisfying the following conditions.

For all , the line intersects .

For all , we have for some satisfying
(1.2)
Note that it is the same direction on both sides of the formula in Theorem 1.1 (ii).
Our second result is the following.
Theorem 1.3.
(Invisible mirror) For every and there exists a set which consists of a finite number of line segments and has the following properties. Let .

For almost all , the light ray arriving from the direction reflects only once in a mirror in and .

For all satisfying we have .
The rest of the paper is organized as follows. We will construct the set described in Theorem 1.1 as the union of a large number of very small fractallike sets. We will call these sets blocks. The construction of an individual block will be based on a Falkoner’s argument and it will be presented in Section 2. We will assemble blocks into a starshaped structure (set ) in Section 3. We will analyze the shadow of in the same section. Section 4 will be devoted to the analysis of light reflections in the set . Section 5 will be devoted to the proof of Theorem 1.3.
We are grateful to Donald Marshall and Boris Solomyak for very helpful advice.
2. Building blocks
Let . The symmetric difference of sets and will be denoted .
The following result is a special case of Falconer’s “digital sundial” theorem. See [Fal03, Thm. 6.9] for an accessible statement of the result and a sketch of the proof. A rigorous proof was given in [Fal86b] in the multidimensional setting.
Proposition 2.1.
For every , there exists a measurable set with the following properties.

For almost all ,

For almost all ,
In words, the projection of in almost every direction is “indistinguishable” from a line segment of unit length, while the projection in almost every direction is “almost” invisible (has measure zero).
The next proposition is the main step in the construction of our basic building block. Although this result is inspired by and very close to, heuristically speaking, Proposition 2.1, it does not follow directly from Proposition 2.1. Our proof will be based on a claim embedded in a proof of a theorem in [Fal86b].
We will call a set a diamond if it is a closed square with sides inclined at the angle to the axes. We will say that is a diamond set if it is a finite union of diamonds. Let denote the square .
Proposition 2.2.
For every and there exists a diamond set which satisfies the following.

There exists such that and for all ,

There exists such that and for all ,
Proof.
The following claim follows from the assertion made in the part of the proof of Theorem 5.1 in [Fal86b] on page 58 and the first paragraph on page 59 (the assertion is proved later in the same proof). For any there exists a countable family of open balls such that , and
(2.1)  
(2.2) 
Let . Then (2.2) implies that
Let . Then
It follows that for any ,
(2.3) 
Similarly, if then for any ,
and
(2.4) 
The sequence of functions is monotone and converges to 0 in view of (2.1). Hence,
and, therefore, there exists such that
(2.5) 
Fix an satisfying this estimate for the rest of the proof.
Let be the collection of all diamonds with diameters of length and vertices in the lattice . Let be the union of all diamonds that belong to and are subsets of . Let . It follows from (2.3) that for any ,
(2.6) 
Let . Then for any , by (2.4),
(2.7) 
The sequence of functions is monotone and converges to . Hence,
This and (2.5) imply that there exists such that
We let . The we can argue as above that
(2.8) 
Fix a satisfying this estimate. Let . We combine (2.7) and (2.8) to see that
(2.9) 
We now let , and . These sets satisfy the proposition in view of (2.6) and (2.9). ∎
Let and . The following corollary follows from Proposition 2.2 by scaling.
Corollary 2.3.
For all , and there exists a diamond set which has the following properties.

There exists such that and for all ,

There exists such that and for all ,
Definition 2.4.
Consider a diamond set satisfying Corollary 2.3 for some , and . The set is a finite union of diamonds . We can label the vertices of the diamond so that they satisfy and , for some and . Let be the union of three closed line segments , and (see Fig. 1).
We let . We will call the set a block.
Remark 2.5.
(i) A block consists of a finite number of bounded line segments with slope .
(ii) The parameters and do not uniquely define the block . We will adopt the following convention. We fix a single block among all blocks with parameters . Then we let for every set of parameters .
(iii) Since and , the diameter of is less than .
(iv) For our arguments, it is irrelevant that the line segments in the definition of a block are created in sets of three from each diamond in a diamond set. We only need a finite family of line segments with slope which satisfies estimates for the size of projections of a block given in Lemma 2.6 below. A set of the type illustrated in Fig. 2
is perfectly acceptable as a “building block” if it satisfies the conditions listed in Lemma 2.6.
Lemma 2.6.
Suppose that , , and is a block.

There exists such that and for all ,

There exists such that and for all ,
3. The sea urchin
Fix an arbitrarily small . Our construction of the set in Theorem 1.1 will have several parameters—real numbers and an integer . Assume that
(3.1)  
(3.2) 
We will make more assumptions later in the proof.
Recall that is a (small) real number and let be a (large) integer divisible by 4. Let for and note that . Let be the closed rectangle with two of its adjacent vertices equal to and , and the other two vertices on the unit circle . Moreover, we require that does not contain ; this uniquely identifies . We let for . The set consists of two thin rectangles with parallel sides; their long sides lie on the same straight lines.
See Fig. 3.
Recall definitions of squares and blocks from Section 2. We let and assume that is so large that . Note that the width of is and the side of is . We choose the values of the parameters so that . Let be the translation which maps to . The translations , , map onto adjacent squares which fill up the rectangle and then extend beyond . Let be the largest such that . Then we let , , , and .
Heuristically speaking, we placed blocks congruent to in long rows in ’s, tightly against each other. The thin rows of blocks form a spiny sea urchin shape (see Fig. 3).
We will prove that the set satisfies Theorem 1.1, provided we choose appropriate values of the parameters of the construction.
3.1. The shadow
Proof of Theorem 1.1 (i).
We parametrize and using polar coordinates as follows, and ; we will suppress in this notation. We will estimate the measure of the set of such that intersects for some values of .
Let be the horizontal line segment extending from the vertical axis to the right hand side half of the boundary of the unit disc, at the level . This line segment contains the lower horizontal side of . Let . Suppose that . It is elementary to see that we can choose so large (and, hence, so small) that for . Moreover, we can choose so small that for every ,
(3.3) 
Consider , where is as in Lemma 2.6, with replaced by . Suppose that . Then may fail to intersect if it crosses to the left of or to the right of . The measure of the set of such that , , and crosses to the left of or to the right of is bounded by . It follows from Lemma 2.6 (i) that the measure of the set of such that crosses to the right of and to the left of , and does not intersect is bounded by . Combining the two estimates, we obtain
We may assume that is so small that . Hence the last expression is bounded from below by .
The same estimate holds for in place of for , by translation invariance. Since , this implies that
We can now make so small that the last estimate and (3.3) yield
Recall that , and . The last inequality implies that the measure of such that , and is bounded above by . According to Lemma 2.6 (i), . It follows that the measure of such that , and is bounded above by . Summing over all intervals of the form and taking into account both and , we obtain the following estimate,
We can now make so small that the right hand side is less than , i.e.,
(3.4) 
We have constructed a set satisfying part (i) of Theorem 1.1. Of course, part (i) is a trivial statement by itself. We will have to show that the same set satisfies part (ii) of the theorem. ∎
4. Light ray reflections
Lemma 4.1.
Recall that is fixed. There exists a set such that and if , , and satisfies then
(4.1) 
Proof.
Let be as in Lemma 2.6 (ii), with replaced by . Let
Then
(4.2) 
by Lemma 2.6 (ii) and (3.2). Let be the set of all such that if satisfies then . It follows from Definition 1.1 of and (4.2) that . It remains to prove (4.1).
Recall the integer used in the construction of . It is elementary to check that if is large then . Hence, the number of block images inside is bounded by . Therefore, the number of (rotated) block images in is bounded by . According to Lemma 2.6 (ii) we have for . Hence, we obtain using (3.1),
(4.3) 
for .
We will now describe the path of a light ray reflecting from mirrors in in general terms. Let be as in Lemma 4.1. Suppose that the light ray arrives along the line with , , and . According to Lemma 4.1, this ray is very unlikely to hit before hitting . So let us suppose that it did not hit before hitting .
The set consists of a finite number of line segments with slope . The light ray may reflect from a number of them. After each reflection, it will move along a line with equal either to or . It is clear that after a finite number of reflections, the light ray will leave the set . We will argue that at the time the light ray leaves , it is very likely to move along a line with .
Next, the light ray will have another chance to reflect from . We will show that the chance that the light ray will hit is very small, once again using Lemma 4.1.
In summary, a typical light ray arriving in the direction with will avoid hitting on the way to , then it will follow a zigzag path inside , and then it will leave the unit disc without hitting on the way out. A similar analysis applies to light rays arriving from other directions.
4.1. Invariance principle for light rays
By abuse of language, we will refer to as a bundle of light rays, although it would be more precise to say the a bundle of light rays consists of all light rays with .
Lemma 4.2.
Recall that and suppose that and . Consider a bundle of parallel light rays. Suppose that all light rays in reflect from a bounded set consisting of a finite number of parallel line segments and then escape as two bundles of light rays and , with , , , , and . Let be the transformation that takes an element of and maps it to the outgoing light ray in . The transformation is onetoone, except for a finite number of lines in for which is not uniquely defined because these light rays encounter endpoints of mirrors on their way.
(i) We have .
(ii) Moreover, and , where , , , and .
Proof.
The claim is obvious if consists of a single line segment. The general statement can be easily proved by induction on the number of line segments in . We leave the details to the reader. ∎
We note parenthetically that Lemma 4.2 is a special case of a well known and more general theorem in the theory of billiards, see [Tab05, Thm. 3.1] or [CM06, Lemma 2.35].
Consider and the bundle of light rays such that and . We can write for some . Let . In the following lemma, we will ignore the set , that is, we will consider the effect of reflections in on the light rays in . After the light rays leave the set , they will form two bundles of parallel light rays and , where , ,