ON MINKOWSKI MEASURABILITY

Transcription

1 ON MINKOWSKI MEASURABILITY F. MENDIVIL AND J. C. SAUNDERS DEPARTMENT OF MATHEMATICS AND STATISTICS ACADIA UNIVERSITY WOLFVILLE, NS CANADA B4P 2R6 Abstract. Two athological roerties of Minkowski content are that countable sets can have ositive content (unlike Hausdorff measures) and the roerty of a set being Minkowski measurable is quite rare. In this aer, we exlore both of these issues. In articular, for each d (0, 2) we give an exlicit construction of a countable Minkowski measurable subset of R 2 of Minkowski dimension d and arbitrary ositive Minkowski content. We also indicate how this construction can be extended to R n, to construct a countable subset with arbitrary ositive Minkowski content of any dimension in (0, n). Furthermore, we give an examle of a strictly increasing C function which takes a Minkowski measurable subset of [0, ] onto a set which is not Minkowski measurable but of the same dimension.. Introduction For a subset A R N and ϵ > 0, the ϵ-dilation of A is the set A ϵ = {x : x a < ϵ for some a A}. For 0 s N, the s-dimensional uer and lower Minkowski contents of A are defined by M s L N (A ϵ ) (A) = lim su ϵ 0 ϵ N s and L N (A ϵ ) M s (A) = lim inf ϵ 0 ϵ N s. In both of these L N is the N-dimensional Lebesgue measure. We comment that often there is a normalization factor, but for our uroses we can leave this out, as we are mainly interested in if these are finite and ositive and not in their exact value. The normalizing constants are necessary to have consistent values for the content if we consider a set A R n instead as A R m, for some m n. The Minkowski contents behave in a similar manner to the Hausdorff measures in that if 0 α < β < γ N and 0 < M β (A) < then M α (A) = and M γ (A) = 0. Clearly the same behaviour is true for M s. If M s (A) = M s (A), then their common value is the s-dimensional Minkowski content and is denoted by M s (A). This haens recisely when the limit M s (A) := lim ϵ 0 L N (A ϵ ) ϵ N s Key words and hrases. Minkowski dimension, Minkowski content, Minkowski measurability. This work was artially suorted by NSERC in the form of a Discovery Grant (FM) and a Summer USRA (JCS). This work is artially based on J.C. Saunders undergraduate thesis.

2 2 F. MENDIVIL AND J.C. SAUNDERS exists. The uer Minkowski dimension of A is defined by d M(A) = inf{s > 0 : M s (A) = 0} = su{s 0 : M s (A) > 0}, with a similar definition for the lower Minkowski dimension. The Minkowski dimensions are also known as the box dimensions in the literature, articularly in the fractals literature. The Minkowski contents are less well-known, as they have some undesirable roerties. For more on Minkowski content, see [, 2, 3]. Definition. We say that a set A R N is Minkowski measurable if 0 < M d (A) = M d (A) < where d is the Minkowski dimension of A. Thus a Minkowski measurable set has ositive and finite Minkowski content at the correct dimension, its Minkowski dimension. One major drawback of the Minkowski contents are that they are not outer measures. In articular, they are not countably subadditive. This leads to the fact that the Minkowski dimensions are also not countably stable, that is, the dimension of a countable union can be strictly greater than the suremum of the dimensions of the individual arts. This athology is one of those roerties that we exlore (and exloit) in our construction of Minkowski measurable countable sets with arbitrary dimension and content (in Sec. 2). It is easy to see that M s (A) = M s (cl(a)) (similarly for lower content), and thus the Minkowski dimensions and contents don t distinguish between a set and its closure. For this reason, we deal only with closed sets. It is also easy to see that if A is unbounded then L N (A ϵ ) = for all ϵ, and thus we deal only with comact sets. For disjoint comact sets A i, however, both the uer and lower content are finitely additive. This is easy to see as eventually the finitely many A ϵ i will be disjoint. Thus for this case, the lower and uer Minkowski dimensions are also finitely stable. The Minkowski contents (and hence dimension) are well-behaved with resect to dilations. That is, if f : R n R n scales all distances by a factor of r, then M s (f(a)) = r s M s (A) with a similar behaviour for M s. Clearly this means that such functions reserve the Minkowski dimensions. However, unlike Hausdorff measures, Minkowski content and measurability is not so well-behaved for more general tyes of smooth functions. In Sec. 3 we give an examle of this in one dimension. Subsets of R For comact subsets of R there is an elegant characterization of both the Minkowski dimensions and which sets are Minkowski measurable. This begins with a nice geometric observation. Take a comact A R. If L (A) > 0, then it is easy to see that A has dimension one and is Minkowski measurable. Thus, we assume that L (A) = 0. We let (α i, β i ) be the bounded oen sets in R \ A, labeled in order of decreasing size, and let a i = β i α i. For simlicity, assume that A [0, i a i]. For a given ϵ > 0, let N ϵ satisfy a Nϵ + 2ϵ a Nϵ. Then we have (see also [4, 3]) () V (ϵ) := L (A ϵ ) = 2(N ϵ + )ϵ + a i. i N ϵ + To see this, we just notice that if the ga (α i, β i ) is small enough (smaller than 2ϵ), then it is entirely contained in A ϵ. Otherwise, A ϵ only contains an interval of length 2ϵ around a given endoint of the ga. This formula () relates the asymtotics of V (ϵ) (and hence the Minkowski contents) to the asymtotics of the sequence of ga lengths, {a i }. In articular, only

3 ON MINKOWSKI MEASURABILITY 3 the ga lengths matter, not the geometric lacement of the gas on the real line. Thus, we seak of a comact subset of R in terms of its sequence of ga lengths, which we always list in non-increasing order. For a i = i, with > fixed, it is a relatively straightforward calculation to show that this leads to a Minkowski dimension (both uer and lower) of /. In fact, with a bit more work, it can be shown that any set with ga lengths i, > will be Minkowski measurable at dimension /. A remarkable theorem of Laidus and Pomerance [5] (and given a new roof in [6]) shows that all Minkowski measurable subsets of R are of this form. More recisely Theorem. [5, 6] Let A R be comact with Lebesgue measure zero and let {a i } be the sequence of ga lengths for A. Then A is Minkowski measurable of dimension d (0, ) iff a i C i /d for some C > 0. Their results give much more, including a relation between the Minkowski content and the constant C (in [5], Laidus and Pomerance were rimarily interested in questions involving the sectrum of the Lalacian and this articular result was a necessary ste in that study). Fix > and set a = i. Since the ga lengths are all that matter, we can construct a countable subset E [0, i a i] with ga lengths {a i } by setting E = { i n a i : n = 0,, 2,..., } { i This set E is Minkowski measurable with dimension / by Theorem and hence 0 < M / (E ) <. Using E we can construct a Minkowski measurable subset of R of any desired content; we simly scale E by the aroriate amount r so that r / M / (E ) is the desired content. Our aim in the next section is to do the same in R N. That is, our aim is to construct a countable subset E R N which has Minkowski dimension d (0, N) and is Minkowski measurable at this dimension. Then by a simle scaling we can obtain any ositive and finite content. a i }. 2. Countable subsets of arbitrary content The situation in R 2 is more comlicated due to the geometry in R 2 ; there is no formula corresonding to () for V (ϵ) in terms of ga lengths and hence no simle characterization of Minkowski measurability. For certain classes of cut-out sets, a similar tye of analysis can be done, but clearly cut-out sets in higher dimensions form a small ortion of the collection of all ossible comact sets. We content ourselves here with roviding for each d (0, 2) a construction of a countable set E d R 2 with 0 < M d (E d ) <. For d (0, ), we could simly mimic the construction in R and lace the oints along a line in R 2 with gas i /d. However, for d [, 2), this construction will clearly not work as the sum of the ga lengths would be infinite. What is necessary is that the countably many oints accumulate to a limit in a more two-dimensional way, so that the sum of the gas can be finite even when d [, 2). This is the hilosohy behind our construction. We give exlicit details for the construction and for the roof of the Minkowski measurability in R 2. The situation in R N is similar and we briefly discuss the situation in R N, but the attern is analogous to that in R 2 so we leave out the details dimensional case. Throughout this construction, we have a fixed > 0. Our countable set E d is constructed as a oint at the origin together with a subset of oints on a countable collection of concentric circles, S n, centered at (0, 0) and

4 4 F. MENDIVIL AND J.C. SAUNDERS with the radius of S n being n. For each of the circles S n, we choose a finite set of equally saced oints on S n to belong to the countable set E d, see Fig.. Notice that in our illustration the oints towards the middle merge together, as we have to render them with ositive size. This is imortant to note since in Ed ϵ each oint will have an ϵ-ball surrounding it and these balls always merge in the center of E d. Figure. Illustration of countable set. More formally, noting that the distance between S n and S n+ is n (n + ), we set (2) K n = 2π ( ) = 2π ( ) arccos (n (n+) ) 2 2(n ) arccos 2 2 ( ( n n+ ) ) 2 to be the number of equally saced oints we choose on S n. For simlicity, we always choose one of the oints to be on the ositive x-axis, which then comletely secifies the locations of all the other oints. This choice of K n results in the sacing between the oints on S n being asymtotically the same as the distance between S n and S n+, and is obtained by solving for the angle which results in a sacing of n (n + ). Having chosen K n equally saced oints on the circle S n, we define E d to be the set containing the origin and all these K n oints for each S n. Clearly E d is countable and comact. We now determine the Minkowski dimension of E d and show that E d has ositive and finite content in this dimension. These results come from a careful estimation of the area of Ed ϵ, and so we roceed to discuss this with a fixed value of ϵ. We first make some elementary but useful observations. In general, for a given ϵ there will be three different tyes of behaviours of the ϵ-balls. The outermost ones will all be disjoint and this will haen for those oints on S n for small n. The innermost ones will all overla and form a dense core around the origin, in fact, aroximately a disk. Finally, those in the middle will artially overla both with other ϵ-balls on the same S n and with those on S n+ and S n. In this middle range, there are uncovered gas left in the annulus between S n and S n+, see Fig. 2. We now determine these three ranges of n for a given ϵ. The first range is the simlest to determine. We simly need 2ϵ to be smaller than the sacing between the oints. This occurs when 2ϵ n (n + ). Estimating with the Mean Value theorem, we obtain that the cutoff is when ( + ( ) + (3) n n + N ϵ =. 2ϵ) 2ϵ

5 ON MINKOWSKI MEASURABILITY 5 Figure 2. Balls just starting to overla. Thus all ϵ-balls for oints on S i with i N ϵ will be disjoint. The area of this ortion of E ϵ d is N ϵ (4) πϵ 2 K i. i= Before we start obtaining an asymtotic estimate of this area, we briefly mention two facts which are very useful to kee in mind. First, if we have an increasing function ϕ(x) then we can use the estimate Nϵ N ϵ ϕ(x) dx ϕ(i) i= Nϵ 2 ϕ(x) dx + ϕ(n ϵ + ). Thus the sum and the integral are asymtotic as long as the term ϕ(n ϵ + ) is of lower order than either of the integrals. This could cause a roblem if N ϵ grows too quickly, but we will see that we won t have this roblem. The second useful observation is that if 0 < α < β then N c x α + c 2 x β dx N c x β dx. This is a rather obvious fact, erhas, but in our situation we use it reeatedly. Many of our estimates are based on Taylor exansions and thus the lower order terms are of a strictly lower olynomial order and so maniulations with sums and integrals are easily justified. The first ste in estimating (4) is to notice that (5) K i 2π by a Taylor exansion of the function (i + ) + π( ) + 2π arccos ( 2 ( ( x)) ) 2) at x = 0. Notice that we only need this exansion for x > 0. This gives that K Nϵ 2π ( + ϵ 2) π( + 3) We will see below that the integral has order ϵ 2 +, so that our integral is indeed asymtotic to the sum.

6 6 F. MENDIVIL AND J.C. SAUNDERS Recalling from (3) that N ϵ (/2ϵ) /(+) we see N ϵ Nϵ πϵ 2 K i πϵ 2 i= 2π ( π2 2 + x dx 2) ϵ Thus, the area of the outermost range is asymtotically of order ϵ 2 2/(+), which would give a Minkowski dimension of 2/( + ). We will see that the other two arts have the same asymtotic order so this is indeed the dimension of E d. Thus we have roved: Lemma. The contribution of the outermost range to the area of Ed ϵ is asymtotic to π 2 ( 2 + ϵ 2) For the innermost range, we need to estimate n so that i n imlies that the entire area between S i and S i+ is covered by the ϵ-balls. We use a conservative estimate, which will make this n erhas larger than it really is. Our estimate is based on aroximating the annulus between S n and S n+ as a thin rectangle (see Fig. 3). We also assume that the centers of the ϵ-disks are searated by the same distance as the thickness, D, of this thin rectangle (also an aroximation). Given this, the area in the rectangle is comletely covered when ϵ = 2D/2 = D/ 2. Since the searation on S n is n (n + ), this then yields an estimate of the cutoff at ( ) + (6) M ϵ =. 2ϵ That is, for n M ϵ all the ϵ-balls on S n will merge together into one larger ball. The area of this ball, the inner area A inner, will thus be ( ) 2 + ( ) + (7) π(mϵ + ϵ) 2 π 2 ϵ ϵ ϵ 2. Thus again we see that the leading term is of order ϵ 2 2 +, so we have: Lemma 2. The contribution of the innermost range to the area of Ed ϵ is asymtotic to ( ) π ϵ Figure 3. Estimating the cutoff for the innermost range. Notice that M ϵ Nϵ, so that the size of the index set for the middle range, that is i in the range N ϵ i M ϵ, increases without bound. This is

7 ON MINKOWSKI MEASURABILITY 7 unfortunate, as if the size of the middle range was always bounded we could ignore it, since it would only account for a vanishingly small roortion of the total area. It is easy to find an uer bound on the area corresonding to this middle range as πn 2 ϵ πm 2 ϵ ) 2π 2 + ( ϵ This is of the same order as the other two arts. In fact, the area corresonding to this middle range turns out to be of this order in ϵ, and thus we must carefully estimate it. This is the most delicate art of the estimation because of the artial overlas. The underlying idea of our aroach is to estimate, for each annulus, the roortion of the area of the annulus which is covered as a function of i in the middle range N ϵ i M ϵ. The annulus is divided into cells and so we do this as both a function of the index of the cell and of the annulus. For a fixed ϵ, this results in a double sum over the cells and the annuli in the range; we argue that as ϵ 0 (so N ϵ ), this double sum converges to an integral in such a way that the Minkowski content exists. To simlify our exlanation, we will give our illustrations and our calculations as if the annulus between S i and S i+ were a thin rectangle. Clearly this is not the case; however, we do this for ease of understanding and clarity. After we deal with this simlified and aroximate situation, we discuss how things change in the actual situation. We will not derive formulas for the actual situation, but our discussion should convince the reader that such formulas do indeed exist. As existence is all we are roving, this is sufficient. By (5), we see that K n (2π/)(n + ) and so if < 2π then K n+ K n +. What this means is that the oints on S n+ are shifted with resect to those on S n, as in Fig. 4, and thus the ϵ-balls overla in many different ways. As n increases, this results in these overlas exhibiting the range of all the ossible tyes of shifts. If 2π, then there is a shift only between some fraction of the transitions from S n to S n+, so most of the overlas are more like those in Fig. 3 than those in Fig. 4. We discuss these two cases searately. Notice that since all the disks are of radius ϵ, the area of intersection of any two disks will deend only on the distance between their centers. In fact, the area of this lens is ( ) ] D ϵ [2 2 ϵ2 D arccos 2 2 /4 D, 2ϵ ϵ 2ϵ where ϵ is the radius and D is the distance between the centers. Figure 4. Two rows of the middle range with the shifts and cells. The case where < 2π

8 8 F. MENDIVIL AND J.C. SAUNDERS First we deal with the case < 2π for which K n+ K n +. We refer to Fig. 5 for illustrations. The height of each cell between S n and S n+ is d = n (n + ) while the width is d = (n + ) (n + 2). We see that ( ) ξ := d d = (n + ) (n + 2) n+2 n (n + ) = ( ) + n n n + 2 = 2 n + 2. For a given sacing d, we see that for n d (/2d) /(+) we have n d (n d + ) d and thus ( ) 2 + ξ 2 d + as the asymtotic ratio d : d. In a similar way we see that with d n = n (n+), then (8) d n n +. Now we examine some of the ossibilities. We give exlicit formula for two of the cases just to indicate what form the formulas take. The exact formulas are not imortant, only the fact that they are well-behaved as a function of their arameters in the range of interest. The variable t reresents the shift in the osition of the oint on S n ; t is measured from the uer left corner and is in the range 0 t d /2. For d large, that is, for an annulus towards the start of the middle range and for a shift t close d /2 (so close to the center of the cell), we have the situation as illustrated in the first image in Fig. 5. In this case, the area which is covered is πϵ 2 ϵ 2 x2 dx ϵ 2 x2 dx t/ϵ (d t)/ϵ ( ) ] ξd ϵ [2 2 ϵ2 d arccos 2 ξ 2 /4 dξ (9). 2ϵ ϵ ϵ The two integral terms correct for the fact that the ends of the uer disk are cutoff since 2ϵ > d > d, as can be seen in the first image in Fig. 5. On the other hand, for any d, if t is close to 0 then we have the situation as illustrated in the third image in Fig. 5. Here, the area which is covered is given by (t+(d d ))/ϵ πϵ 2 ϵ 2 x2 dx t/ϵ ( ) ] d ϵ [2 2 ϵ2 d arccos 2 /4 d 2ϵ ϵ ϵ ( ) ] ξd ϵ [2 2 ϵ2 ξ arccos 2 d 2 /4 ξd 2ϵ ϵ ϵ ( ) ϵ 2 t2 + d 2 arccos 2 ϵ 2 t 2 + d 2 /4 t2 + d (0) 2. 2ϵ ϵ ϵ The integral term is to comensate for the fact that the sacing between the oints on S n is d while the width of the cell is d = ξd < d. Thus, rather than the to disks contributing πϵ 2 /2 they contributes a bit less (minus the overlas, of course). We are rimarily interested in the ratio of the area which is covered, so we divide the above by the area of a cell, d 2 ξ. For a fixed ϵ there are only a finite number of different t values which are reresented in any annulus and there are also only a finite number of different values of the arameter d. In fact, M ϵ N ϵ (2 2+2 )Nϵ different d values and at

9 ON MINKOWSKI MEASURABILITY 9 t d d d d t t d d d d Figure 5. Illustration of tyes of overlas for middle range. most K i different t values on a given annulus. As ϵ 0, both d and t assume more values until, in the limit, they fill out their range. The range of t is simly 0 t d /2 = ξd/2. The range of d is obtained by noting that the middle range corresonds to those annuli whose width is between 2ϵ and 2ϵ and thus 2ϵ d 2ϵ. Notice that everything scales with ϵ. If we reduce ϵ by some factor ρ, then the middle range is shifted and both the arameters d and t simly scale exactly by a factor of ρ. That is, there are cells in this new range which corresond to scaled versions of the cells in the old range. This oint is crucial in showing convergence as this is what leads to being able to aroximate the double sum with an integral. It also means that we can scale out the deendence on ϵ and obtain the limiting ranges () 2 d 2, and 0 t d /2. We now describe the function ϕ(d, t). This function gives the roortion of the area in a cell which is covered, with width arameter d and shift t. We could give an exlicit exression for ϕ, but the secific details are not very imortant (esecially since, at the moment, we are only describing the aroximation where we think of the annuli as rectangles). What is imortant is to understand that such an exression does exist and that the function is a differentiable function of its arguments. The differentiability can be seen either from geometrical considerations or by noting that the formula is iecewise defined by exressions of the form (9) and (0) and these exressions are differentiable.

10 0 F. MENDIVIL AND J.C. SAUNDERS For 2 d 2/ + ξ 2 /4 there are only intersections of the tye illustrated in Fig. 5 images a) and c). If 0 t 4 d 2, then the intersection is of the tye illustrated in Fig. 5 image c). If 4 d 2 < t d /2, then we have the situation illustrated in image a) of that figure. Thus, for these ranges of d and t, we have use these figures to derive exressions for ϕ. If 2/ + ξ 2 /4 < d 2, then we have intersections of the tye illustrated in Fig. 5 images c) and d). For 0 t d 4 d 2 we have the situation from Fig. 5 c) and for d 4 d 2 < t d /2 we have the situation illustrated in d). The resulting iecewise exression for ϕ(d, t) is rather long and messy, but comosed of differentiable functions which stitch together differentiably at the boundaries. Furthermore, since we can scale the ϵ deendence out, ϕ(d/ϵ, t) reresents a dimensionless roortion of the area of a cell which is covered. The roortion is always strictly ositive; that is, there is some η > 0 so that (2) 0 < η ϕ(d/ϵ, t) for all t, d. In the end, for a fixed ϵ we obtain an exression of the form (3) ξ i d 2 i ϕ(d i /ϵ, t j ). N ϵ i M ϵ j over the cells The inner sum is easily converted to one over the various shifts, t i. In fact, for the annulus between S n and S n+, we have rational shifts of the form l/(k n K n+ ) for various values of l and these uniformly and densely fill the interval [0, d /2] as ϵ 0. This is a result of the asymtotic estimate K n (2π/)(n + ) and so K n+ K n (2π/). We will transform the outer sum into an integral over the values of d in the range 2 d 2 by a suitable change of variable. Notice that we have to be careful since the values of d i are not uniformly distributed over this range, as there are more small values of d i than large values. However, the change of variable will naturally account for this. Returning to our sum (3), we first notice that the inner sum can be written as ϕ(d i /ϵ, t j ) = K i ϕ(di /ϵ), j over the cells where ϕ is the average value of ϕ over this annulus. This average value only deends on the quantity d/ϵ. Continuing, we use the asymtotic exressions for d i, ξ, and K i to obtain (after some simlification) (4) N ϵ M ϵ i=n ϵ [ i ] 2π i 2+ ϕ i. Now, using an integral aroximation, we get that this is asymtotic to ] Mϵ (5) [ π x x ϕ 2+ x dx. We now use the change of variables y = (/ϵ)x = d/ϵ. The reason for this change of variable is that then x = N ϵ becomes y = 2 and x = M ϵ becomes y = 2. After this change of variable, our integral is (6) 2π + ( ϵ ) y=2 y= 2 [ ( ϵ ) + y + ] y 2 + Φ(y) dy.

11 ON MINKOWSKI MEASURABILITY As we have no secific form for ϕ(x), we don t have a secific form for Φ(y). However, ϕ is only a function of d/ϵ = y, so Φ is only a function of y. Further, our bound (2) gives 0 < η Φ(y), so that the value of the integral is bounded away from zero. Thus, as ϵ tends to zero, the integral converges to the constant value (7) C := y=2 y= 2 y 2 + Φ(y) dy and so the area of this middle range is asymtotically equal to (8) C 2π ϵ 2 2 +, in articular it is of asymtotic order ϵ 2 2 +, as desired. Lemma 3. In the case that < 2π, the contribution of the middle range to the area of Ed ϵ is asymtotically equal to where C is as given in (7). 2π 2 + C ϵ 2 2 +, + The case where 2π If 2π, then much of the time K n = K n+ and so we don t get the same tyes of shifts between the oints on successive S n. In this case, we can do a similar analysis as above, excet for most of the annuli the situation is as illustrated in Fig. 3 and only for a small roortion are there shifts. We briefly discuss the differences that this introduces in the analysis. Because of the two different behaviours, we will have two different functions: ϕ(d/ϵ, t) for the shifts and ψ(d/ϵ, t) for the situation where there are no shifts. As before (in equation (3)), we sum over the annuli and cells and obtain a sum ξ i d 2 i ϕ(d i /ϵ, t j ) N ϵ i M ϵ j over shift cells (9) + ξ i d 2 i ψ(d i /ϵ, t j ). N ϵ i M ϵ j over no-shift cells We again can relace the sum over the cells with an average value of ϕ and ψ resectively, and substitute asymtotic exressions to obtain ] ] (20) [ π i i ϕ 2+ i + [ π i i ψ 2+ i. Nϵ i Mϵ Nϵ i Nϵ shift cell no shift cell Now we wish to estimate this as an integral. We define S ϵ and N ϵ as subsets of [N ϵ, M ϵ ] given by S ϵ = {[i, i + ] : the annulus between S i and S i+ has shifts } and N ϵ = [N ϵ, M ϵ ] \ S ϵ, so will also be union of disjoint intervals. Thus, asymtotically we can estimate the sums in equation (20) as the sum of the integrals ] [ 2 +2 ] + [ S ϵ x 2π x 2+ ϕ x dx + N ϵ x 2π x 2+ ψ x dx. Using the same change of variables as before, y = (/ϵ)x = d/ϵ, the two sets S ϵ, N ϵ are transformed into two subsets of Ŝϵ, N ϵ [ 2, 2]. Again these subsets are

12 2 F. MENDIVIL AND J.C. SAUNDERS unions of intervals and we obtain the sum of two integrals, each one as in (6), one over Ŝϵ and one over N ϵ. We claim that as ϵ 0, both of these integrals converge. The key is to understand what haens to Ŝϵ and N ϵ as ϵ 0. S ϵ consists of aroximately 2π(M ϵ N ϵ )/ intervals each of unit length and evenly distributed in the interval [N ϵ, M ϵ ] and searated by a distance of /(2π). This means that Ŝ ϵ also consists of this number of disjoint intervals and also distributed throughout the interval [ 2, 2]. As ϵ 0, the set Ŝϵ becomes dense in [ 2, 2] (as does N ϵ ). Thus the integral of any continuous function g over Ŝϵ will limit to some integral of g over the entire interval [ 2, 2]. The intervals of S ϵ are uniformly distributed in [N ϵ, M ϵ ] with density 2π/. This means that Ŝϵ will asymtotically be uniformly distributed in [ 2, 2] with density 2π/ as well. Therefore, for any continuous g : [ 2, 2] R, we have that and Thus, we have: lim ϵ 0 Ŝ ϵ g(y) dy = 2π 2 2 g(y) dy, ( lim g(y) dy = 2π ) 2 ϵ 0 g(y) dy. N ϵ 2 Lemma 4. In the case > 2π, the sum contribution of the middle range to the area of Ed ϵ is asymtotic to 2π 2 + C ϵ 2 2 +, + where this time C reresents the weighted sum of the two integrals, one for Φ and one for Ψ. Finally, recall that, for clarity, our discussion was restricted to assuming that the annuli are rectangles and not actual annuli. What changes if instead of develoing formulas for ϕ based on thin rectangles we develo similar formulas for ϕ based on the actual situation of annuli? Clearly the formulas themselves will be different, but only in the details, not in the general form. In fact, the overall situation changes very little. It is for this reason that we illustrated with rectangles rather than with annuli. The conclusion is still the same. Thus, utting together lemmas, 2 and either lemma 3 or lemma 4, we get the following theorem. Theorem 2. Let d (0, 2) and C > 0 be given. Then there is a countable set E d R 2, of the form described above, so that the Minkowski dimension of E d is d, the set E d is Minkowski measurable, and the Minkowski content of E d is exactly C N-dimensional case. The same construction can easily be erformed in R N and the details of the estimation extended. For any > 0, we set S n to be a shere of radius n centered at 0 and we then choose oints on S n whose minimum searation is asymtotically n (n + ). Now, it is not so simle to choose oints on a shere in R N which are all uniformly laced and uniformly saced (usually it is imossible to do this). In fact, in general, it is a difficult comutational roblem to try to even aroximately do this (see [7, 8]). However, we don t need to have the sacing comletely uniform. All we really need is that the sacing is asymtotically within a constant multile of n (n + ) and that we can do the same tye of scaling that we did in the examle in R 2, above.

13 ON MINKOWSKI MEASURABILITY 3 If the oints were to be laced in R N instead of on the shere, we would lace them on a scaled version of the integer lattice, where the oints are saced on hyercubes of side length n (n + ). This would result in a range of interoint sacing from n (n + ) to N (n (n + ) ) for the diagonal distance. As this structure scales nicely, similar arguments to those above would show convergence. For our lacement on the shere of radius n in R N, we use the olar coordinates subdivision method [8]. The rocedure recurses down the dimensions. To describe this method, let x i be the generalized sherical coordinates, that is i x i = ρ cos(ϕ j ) sin(ϕ i ), i N 2, and j= N 2 x N = ρ cos(ϕ j ) cos(ϕ N ), j= with ϕ i [0, π) for i =,..., N 2 and ϕ N [0, 2π). We set ρ = n for the shere S n. With = n (n + ), the desired sacing between the oints, we choose a grid of πn / equally saced oints in [0, π) for the grid in the ϕ direction. When we fix ϕ to any of these values, we have an N 2 dimensional shere whose radius is n cos(ϕ ). We recurse by choosing a grid of πn cos(ϕ )/ equally saced oints in [0, π) for the grid in the ϕ 2 direction on this articular N 2 dimensional sherical slice. Asymtotically, this results in 2π N/2 n N Surface area of shere of radius n Γ(N/2)(n (n + ) = ) N hyervolume of cell in square grid oints on the shere of radius n, which is just what one would exect for a hyersquare grid in R N of side length n (n + ) in a region with total volume equal to the surface area of the given shere. The estimation of the volume of the outer and inner ranges are more-or-less exactly analogous as in R 2, the only change being that the inner range doesn t start until there is comlete overla for the diagonal oints on S n and S n+. This at most multilies M ϵ by a constant factor of N /(2+2) ; the starting oint of the middle range remains unchanged. For the middle range, the estimate will involve the same reasoning as in R 2, the formulas being much more involved but the logic remaining the same. The Minkowski dimension of the resulting countable subset of R N is N/( + ) and the content is ositive and finite. With this, we can obtain the following: Theorem 3. Let d (0, N) and C > 0 be given. Then there is a countable set E d R N, of the form described above, so that the Minkowski dimension of E d is d, the set E d is Minkowski measurable, and the Minkowski content of E d is exactly C Alternative construction for dilations using the l norm. If we are willing to modify the definition of the dilation of a set by using a slightly different distance, the above construction can be reeated and the analysis greatly simlified. For this section, we will use the definition A ϵ = {y : a y < ϵ, for some a A}, so that we use the suremum norm rather than the usual Euclidean distance. We again obtain the same result, that for any d (0, N) and any C > 0 there is a set

14 4 F. MENDIVIL AND J.C. SAUNDERS E d R N with Minkowski dimension d and with Minkowski content exactly equal to C. It is not hard to see that comuting the Minkowski dimension with either definition of dilation will yeild the same result. However, it is not clear if both dilations give the same class of Minkowski measurable sets it is likely that they do not. This time the construction uses concentric hyercubes rather than concentric sheres (see Fig. 6). The hyercube S n is centered at the origin and has side length 2n, just as in the revious construction. We again lace finitely many oints on S n. For each N dimensional face, we lace the oints along coordinate directions with a searation of n ( / + ) (which is asymtotic to n (n + ) ). This gives, asymtotically, a total of (2n( / + ) + ) n oints on each face. The major simlification is that, with the new definition of an ϵ-dilation of a set, the geometry is much simler. For instance, once the ϵ-dilation of oints on S n intersect, the union of these ϵ-squares will form a rectangular tubular neighborhood of the entire square S n and this area is simle to comute. Furthermore, either the tubular neighborhood of S n will join with that of S n+, comletely filling the region inbetween, or they don t overla at all. Thus we don t have the comlication of all the different tyes of overla. It is also clearly much simler to arrange oints on the faces of these squares than on the shere, and this substantially simlifies the analysis. We again use Lebesgue measure to measure the volume of Ed ϵ (with the new definition of ϵ dilation). We obtain the following result: Theorem 4. Let E R N be the countable subset described above. Then using the l ϵ-dilation, we have that the Minkowski dimension of E is N/( + ) and the Minkowski content is 2 2N+N + where a = / +. ) (a N + N N+N + N( N+ + a N+ N + + ) Notice that we have the same Minkowski dimension as the set constructed with sheres. This is not a surrise as it is ossible to show that there is a bi-lischitz maing from one set onto the other, so their dimensions must agree., Figure 6. Illustration of squares and square dilations.

15 ON MINKOWSKI MEASURABILITY 5 3. A strictly increasing C function which does not reserve Minkowski measurability Clearly any linear function f : R R will reserve Minkowski measurability and simly scale the Minkowski content. It is also easy to see that any bi-lischitz function will reserve the Minkowski dimension. However, it is not so clear what are necessary and sufficient conditions on a function f to reserve Minkowski measurability. The following examle is quite surrising in that continuity in the local distortion is not sufficient to guarantee that a function reserves Minkowski measurability. This means that a function has to satisfy quite strong conditions in order to guarantee that it reserves Minkowski measurability. Contrast this to the case of Hausdorff measures, where if 0 < H d (A) < and f is bi-lischitz then 0 < H d (f(a)) <. We first start with a lemma which describes a general construction. Since the Minkowski dimension, measurability, and content (if it exists) of a comact subset of R deends only on the sequence of ga lengths, this construction starts with these ga lengths and shows how to construct a whole family of comact sets with exactly these ga lengths. The idea is similar to that given in [9, 0]. If is a total order on a set S, a cut is a artition {L, R} with L R = S and for all l L and r R we have l r. Note that we allow either L or R to be emty. Lemma 5. Let a n be a ositive summable sequence. Let be any total order on N. Define the set C a = { i S a i : (S, S 2 ) a cut of N in the order }. Then C a is a closed subset of [0, n a n] with Lebesgue measure zero. Proof. Let I = [0, n a n]. The two oints x n = a n + i n a i and x n = i n a i are both in C a. By the definition of C a, there are no oints y C a with x n < y < x n. Since x n x n = a n, we see that I \ C a contains an oen interval of length a n, namely the interval (x n, x n ). Since this is true for each n and all these intervals are disjoint, I \ C a n a n = I and so C a = 0. To show that C a is closed, suose that x n x where x n C a. Let (S n, S2 n ) be the cut of N associated with x n and define S = n S n and z = i S a i. We will show that x = z. First, by definition we see that x n z for all n and so x z. Let ϵ > 0 be given and choose N so that n N imlies that i n a i < ϵ. Now, there is some M so that for any n M we have S \ S n {i N} so for such n we have i S \S a n i < ϵ. However, this means that z x n < ϵ. The roof for x n x is similar. Since any convergent sequence x n x contains a monotone subsequence, we are done. We now describe our construction of the romised function. Let a n = /3 /n s where s = log(3)/ log(2). Then we can also write a n = 3 log(n)/ log(2). We also consider b n = 3 log(n)/ log(2) so that b n = /3, /9, /9, /27, /27, /27,.... That is, b n is the set of ga lengths of the standard middle-/3 Cantor set. From Theorem we see that any closed set with comlementary ga lengths a n is Minkowski measurable and any closed set with comlementary ga lengths b n is not Minkowski measurable. The nonmeasurability of the classical Cantor set was noticed in [5] and is a consequence of the oscillations of V (ϵ). This can be seen

16 6 F. MENDIVIL AND J.C. SAUNDERS exlicitly in Eq. (.) of [3], for examle. We construct a C function which takes C a bijectively onto C b. We see that (2) c n := b n = 3 log(n)/ log(2) log(n)/ log(2) frac(log(n)/ log(2)) = 3 a n satisfies c n 3. First we note that {c n } is dense in [, 3]. 0 i < 2 k we have To see this, for n = 2 k + i with log(n)/ log(2) = log 2 (2 k + i) = k + log 2 ( + i/2 k ). Since numbers of the form + i/2 k are dense in [, 2], log 2 ( + i/2 k ) is dense in [0, ] and thus c n is dense in [, 3]. Define the total order on N by n m if either c n < c m or if c n = c m but n < m. Using this order, we define C a and C b as in the lemma. Further, we use the notation I a = [0, n a n] and similarly for I b. Notice that if (x n, x n ) and (x m, x m ) are two comlementary intervals for C a with x n < x m then c n c m. This fact is imortant. We define the function f : I a I b by first defining f : C a C b and extending. For x C a, we see that there is some cut (S, S 2 ) of N with x = i S a i. So, using this we define (22) f(x) = i S b i. Now we extend f to I a by linearly extending it to each comlementary interval. That is, for t (x n, x n ) we define f(t) = f( x n) f(x n ) (t x x n x n ) + f(x n ) = c n (t x n ) + f(x n ) = c n (t x n ) + b i. n i n Notice that the sloe of f on (x n, x n ) is c n. Proosition. The function f is continuously differentiable. Proof. We show this by exlicitly exhibiting the derivative of f, which we will call g. First we define g on I a \ C a by g(x) = c n for any x (x n, x n ). We claim that we can continuously extend g to all of I a. To do this, we define g on C a in such a way to make it continuous. For x = i S a i, where (S, S 2 ) is the defining cut for x C a, we let (23) g(x) = su{c n : n S }. Notice that g is increasing on C a and constant on each comonent of I a \ C a. In fact, g is very much analogous to the Cantor ternary function. Let x C a be defined by the cut (S, S 2 ). Now, suose that y nk (x nk, x nk ) with y nk x. This means that n k S and thus c nk g(x). Suose that there is some d with c nk d < g(x). As {c i } is dense in [, 3], there is an M with c nk < c M < g(x) which imlies that y nk x M < x M x and so x y nk a M > 0, a contradiction. Thus c nk g(x). In a similar way we can see that if y nk x with y nk (x nk, x nk ) then c nk g(x). To finish the roof that g is continuous, we just note that if y C a with y = i S a i, then for any n S and m S 2, we have c n g(y) c m. This imlies that it is enough to only consider sequences from the gas which converge to x.

17 ON MINKOWSKI MEASURABILITY 7 Next we show that f (x) = g(x) for all x in the interior of I a. If x I a \ C a, then x (x n, x n ) for some n and thus x = x n + (x x n ) so f(x) = b m +c n (x x n ) = a n c m +c n (x x n ) = xm x g(t) dt+ g(t) dt = m n n m m n x m x n On the other hand, if x = i S a i, then we have f(x) = i S i b i = i S a i c i = i S xi x i g(t) dt = x 0 g(t) dt. Notice that g(t) 3 for all t, and thus f (t) 3 for all t I a. 4. Closing comments and further questions The situation in one dimension is very nicely characterized by the revious work in [5, 6]. It would be interesting to have such a characterization for higher dimensions (even two dimensions). However, it is likely that such a characterization is not ossible or at least not such a nice characterization. Perhas it might be ossible for countable sets. In our construction, we use the simlest tye of countable comact set in that our sets all have only one limit oint. This means that all other oints have only finitely many nearest neighbours, which simlifies the situation. A characterization of the Minkowski dimension and measurability of an arbitrary countable comact set would have to contend with erhas infinitely many limit oints. There is a nice inductive characterization of countable comact sets given in a classic aer by Mazurkeiwicz and Sierinski []. One aroach might be based on a transfinite induction on the tye of the countable comact set. A further direction to exlore is to determine the recise roerties that a function f must satisfy in order to reserve Minkowski measurability. This is not a simle task even for subsets of R, as our examle indicates. One would think that having the local distortion change in a continuous fashion would be enough, but our examle shows that it is not a strong enough condition. References [] H. Federer, Geometric Measure Theory, Sringer-Verlag, New York, 969. [2] P. Mattila, Geometry of sets and measures in Euclidean saces, Cambridge University Press, Cambridge, 995. [3] M. L. Laidus and M. van Frankenhuijsen, Fractal Geometry, Comlex Dimensions and Zeta Functions: Geometry and Sectra of Fractal Strings, Sringer, New York, [4] K. Falconer, Techniques in Fractal Geometry, John Wiley, 997. [5] M. L. Laidus and C. Pomerance, The Riemann zeta-function and the one-dimensional Weyl-Berry conjecture for fractal drums, Proc. London Math. Soc. (3) 66, no (993), [6] K. J. Falconer, On the Minkowski measurability of fractals, Proc. Amer. Math. Soc. 23, no. 4, (995), [7] E.B. Saff and A.B.J. Kuijlaars, Distributing many oins on a shere, Math. Intelligencer 9, no. (997), 5 -. [8] A. Katanforoush and M. Shahshahani, Distributing oints on the shere, I., Exeriment. Math. 2, no. 2, (2003), [9] A. Besicovitch and S. Taylor, On the comlementary intervals of a linear closed set of zero Lebesgue measure, J. London Math. Soc. 29 (954) [0] C. Cabrelli, F. Mendivil, U. Molter, R. Shonkwiler, On the Hausdorff h-measure of Cantor Sets, Pacific Journal of Mathematics, 27()45-59, [] S. Mazurkiewicz and W. Sierinski, Contribution à la toologie des ensembles denombrables, Fund. Math., 7-27, 920. x 0 g(t) dt.