Some results in support of the Kakeya Conjecture

Transcription

1 Some results in support of the Kakeya Conjecture Jonathan M. Fraser School of Mathematics, The University of Manchester, Manchester, M13 9PL, UK. Eric J. Olson Department of Mathematics/084, University of Nevada, Reno, NV 89557, USA. James C. Robinson Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK. Abstract A Kakeya set is a subset of R d that contains a unit line segment in every direction. We introduce a technique called cut-and-move designed to study Kakeya sets and use this to make several simple observations in support of the famous Kakeya conjecture. For example, we give a very simple proof that the lower box-counting dimension of any Kakeya set is at least d/2. We also examine the generic validity of the Kakeya conjecture in the following sense. Let S d 1 denote the unit sphere in R d with antipodal points identified. We encode a Kakeya set in R d as a bounded map f : S d 1 R d, where f(x) gives the centre of the unit line segment orientated in the x direction. Denoting by B( S d 1 ) the collection of all such maps equipped with the supremum norm, we show that (i) for a dense set of f the corresponding Kakeya set has positive Lebesgue measure and (ii) the set of those f for which the corresponding Kakeya set has maximal upper box-counting (Minkowski) dimension d is a residual subset of B( S d 1 ). Mathematics Subject Classification 2010: 28A80, 54E52, 28A75. Key words and phrases: Kakeya conjecture, box-counting dimension, Hausdorff dimension, Baire category. addresses: jon.fraser32@gmail.com (Jonathan M. Fraser), ejolson@unr.edu (Eric J. Olson), j.c.robinson@warwick.ac.uk (James C. Robinson) 1

2 1. Introduction A Kakeya set in R d is a subset of R d that contains a unit line segment in every direction. For convenience (and because most of our results concern box-counting dimension which is usually only defined for bounded sets) we will assume that our Kakeya sets are bounded. In 1919 Besicovitch proved the surprising result that there are Kakeya sets that have zero measure, see the later 1928 paper [1]. Given this, it is natural to ask whether the sets can in fact be even smaller, i.e. whether there are Kakeya sets with zero s-dimensional Hausdorff measure for some s < d. The Kakeya Conjecture is that this is not possible: equivalently, that the Hausdorff dimension of any Kakeya set in R d is maximal, i.e. equal to d. Davies [3] showed that this conjecture is true when d = 2. Wolff [10] showed that dim H (K) (d+2)/2, and Katz & Tao [5] showed that dim H (K) (2 2)(d 4) + 3. With the Hausdorff dimension replaced by the upper box-counting dimension, the best bound in three dimensions is now dim B (K) 5/2 + ϵ; the hard-won improvement over the result of Wolff is due to Katz, Laba, & Tao [6]. In this paper we provide several results in support of the Kakeya conjecture. Our main tool is the simple observation that one can decompose and translate a set without altering its dimension, a process we call cut-andmove. Using this method we also give a very simple proof that the lower box-counting dimension of any Kakeya set is at least d/2. We also show that the box-counting dimension version of the Kakeya Conjecture is true generically, in the following sense. Let S d 1 denote the unit sphere in R d with antipodal points identified. We encode a Kakeya set in R d as a bounded map f : S d 1 R d, where f(x) gives the centre of the unit line segment orientated in the x direction. Denoting by B( S d 1 ) the collection of all such maps equipped with the supremum norm, we show that (i) for a dense set of f the corresponding Kakeya set has positive Lebesgue measure, in contrast to Besicovitch s existence result mentioned above, and (ii) the set of those f for which the corresponding Kakeya set has maximal upper box-counting (Minkowski) dimension d is a residual subset of B( S d 1 ). The Baire Category Theorem has previously been used by Körner [7] in the context of the Kakeya problem to prove the existence of zero measure Kakeya sets ( Besicovitch sets ). 2

3 2. Notions of dimension The Kakeya Conjecture, that the dimension of any Kakeya set in R d is maximal, i.e. d, comes in (at least) two flavours. The strong version uses the Hausdorff dimension, the weaker the upper box-counting dimension. To define the Hausdorff dimension, we first define the s-dimensional Hausdorff measure of a set A as { H s (A) = lim inf U j s : A } U j, U j r, r 0 j j where U denotes the diameter of the set U. Then dim H (A) = inf{s : H s (A) = 0}. When s is an integer, H s is proportional to s-dimensional Lebesgue measure in R s [4, 9]. The upper box-counting dimension is defined for any bounded set A, and has a variety of definitions, the most common being dim B (A) = lim sup ϵ 0 log N(A, ϵ), (2.1) log ϵ where N(A, ϵ) denotes the minimum number of balls of radius ϵ required to cover A. Note that it is an immediate consequence of the definition that dim B (A) = dim B (A) (where A denotes the closure of A) in contrast to the Hausdorff dimension, which does not have this property. We will use an equivalent definition in Section 6. This dimension is also referred to as the (upper) Minkowski dimension. Replacing the lim sup in (6.1) by a lim inf we obtain the lower boxcounting dimension: dim LB (A) = lim inf ϵ 0 log N(A, ϵ). (2.2) log ϵ It is well known that in general dim H (A) dim LB (A) dim B (A) for any bounded set A [4, 9]. 3

4 3. The cut-and-move technique Our first tool is the following simple cut-and-move (decomposition and shifting) lemma, which works for any dimension that is monotonic, stable under finite unions, and translation invariant. A dimension is monotonic if stable under finite unions if ( n dim A B dim(a) dim(b), and invariant under translations if ) A j = max dim(a j ), (3.1) j dim(t a A) = dim(a), where T a A = {x a : x A}. These properties are enjoyed by all the common dimensions: Hausdorff, packing, upper-box (Minkowski), and Assouad (see Falconer [4] or Robinson [9] for a survey of dimensions). Among these, the Hausdorff and packing dimensions are also stable under countable unions (take a countable union in (3.1), replacing the max by a sup on the right-hand side). We denote by B(R d ) the collection of all bounded subsets of R d. Lemma 3.1. Suppose that dim: B(R d ) [0, ) is monotonic, stable under finite unions, and translation invariant. Let K be any bounded subset of R d with n K = K j, (3.2) where the union need not be disjoint. Then for any {a j } n R d, ( n ) dim(k) = dim T aj K j. (3.3) If dim is stable under countable unions then one can allow countable unions in (3.2) and (3.3). 4

5 Proof. Since K j K and dim is monotonic, dim(k j ) dim(k). Since dim is stable under finite unions, dim(k) = max j dim(k j ). Since the dimension is unaffected by translations, dim(t aj K j ) = dim(k j ). It follows that dim(k) = max j dim(k j ) = max dim(t aj K j ) = dim ( ) n T aj K j. j A fairly simple application of this procedure yields the following result, in which dim can be taken to be any dimension satisfying the conditions of Lemma 3.1, or the lower box-counting dimension 1. It says that we can move all the line segments in any Kakeya set to within an arbitrary small distance of the origin without altering its dimension. By keeping track of the number of elements in the cover of K as we apply the cut-and-move procedure we will obtain a bound on the lower box-counting dimension of any Kakeya set in Theorem 4.1, below. Lemma 3.2. Given a Kakeya set K, for any ϵ > 0 there exists another Kakeya set ˆK such that dim( ˆK) = dim(k) and ˆK consists of unit line segments whose centres lie within ϵ of the origin. Proof. Choose M > 0 such that K is contained in ( M, M) d, and cover the set ( M, M) d with a disjoint collection of d-dimensional cubes d i=1 [x i, x i) with sides of length M/n for some integer n such that M/n < ϵ/ d; denote this family of cubes by {Q j } N, and their centres by a j. If K j is taken to be the set of unit line segments whose centers lie in Q j then (3.2) holds, and hence using Lemma 3.1 the set has the properties required. ˆK = N T aj K j Figure 1 shows the procedure used in the proof of Lemma 3.2 applied to a discrete Kakeya set consisting of 64 line segments, cutting the original two-dimensional set into 4 n portions and moving the centres of the resulting squares to the origin. 1 The lower box-counting dimension is not stable under finite unions, but the conclusion of Lemma 3.1 is still valid, since N(K, δ) n N(K j, δ) nn(k, δ) for any δ > 0. 5

6 Figure 1: The cut-and-move procedure of Lemma 3.1, as used in the proof of Lemma 3.2, applied to a division of the original set (top left) into 4 n portions, with n = 1 (top right), n = 2 (bottom left), n = 4 (bottom right). Centres (marked with dots) lie within the squares of sides 4, 2, 1, and 1/4, respectively. Since the Hausdorff dimension is stable under countable unions, the following improvement holds for Hausdorff dimension. Lemma 3.3. Given a Kakeya set K, there exists a set K such that dim H ( K) = dim H (K) and for every ϵ > 0 there is a unit line segment in K in every direction whose centre lies within ϵ of the origin. Proof. Let K j be the set ˆK from Lemma 3.2 constructed with ϵ = 2 j, and set K = j K j. Of course, this result does not say anything precise about the Hausdorff dimension of Kakeya sets, but we find it a useful heuristic that provides a strong indication of the plausibility of the conjecture in a perhaps striking way. If one could only push this result a little further to yield a set with a line segment in every direction whose centre was the origin (i.e. the ball of radius 1/2) then this would of course yield a proof of the Kakeya Conjecture. We can achieve this by allowing a small perturbation of our original set, which is the idea behind the density result of Section 5. 6

7 4. The lower box-counting dimension of Kakeya sets Using the cut-and-move procedure from Lemma 3.1 we can now give a very short and simple proof that dim LB (K) d/2; the argument is similar to that of Lemma 3.2. Theorem 4.1. If K is a Kakeya set in R d then dim LB (K) d/2. Proof. If dim LB (K) = s then for any s > s there exists a sequence ϵ j 0 such that K intersects a collection Q j of at most ϵ s j cubes in a partition of R d into ϵ j -cubes. It follows that (i) the set of midpoints of the line segments in K lies within this collection of ϵ j -cubes and (ii) K can be covered by ϵ s j balls of radius dϵ j (since every ϵ-cube lies within a dϵ-ball). We now use the cut-and-move procedure, decomposing K into a collection of at most ϵ s j sets consisting of line segments whose midpoints lie in the ϵ j - cubes in Q j (these sets need not be disjoint). We translate each set in this decomposition so that the centre of the corresponding ϵ j -cube moves to the origin, and we also translate a copy of the entire cover of K by dϵ s j -balls. This yields a set ˆK which can be covered by ϵ 2s j balls of radius dϵ j, and which contains a line segment in every direction whose midpoints lie within ( d/2)ϵ j of the origin. Now simply observe that any point in B(0, 1/2) lies within ϵ j of a point in ˆK, so by expanding the radius of the balls in the cover of ˆK to 2 dϵ j we obtain a cover of B(0, 1/2); it follows that N(B(0, 1/2), 2 dϵ j ) ϵ 2s j. Since the lower box-counting dimension of B(0, 1/2) is d, we must have 2s d, and since this holds for any s > s, it follows that s d/2 as claimed. While there are much stronger results available, we think that the simplicity of the argument above is appealing in its own right. 5. Density of Kakeya sets with positive measure Let S d 1 denote the unit sphere in R d with antipodal points identified. We encode a Kakeya set in R d as a bounded map f : S d 1 R d, where f(x) gives the centre of the unit line segment orientated in the x direction. We denote by B( S d 1 ) the collection of all such maps, and make this a Banach space by equipping it with the supremum norm. 7

8 Given such a map f, we define K(f) to be the Kakeya set encoded by f, explicitly K(f) = f(x) + tx. x S d 1 t [ 1/2,1/2] We denote by K (R d ) the collection of all compact subsets of R d. If we define ρ(a, B) = sup inf a b, a A b B then the symmetric Hausdorff distance (which yields a metric on the set of all compact subsets of R d ) is given by dist H (A, B) = max(ρ(a, B), ρ(b, A)). We make the following simple observation. Lemma 5.1. If f, f 0 B( S d 1 ) then ) dist H (K(f), K(f 0 ) f(x) f 0 (x). Proof. Given f 0, f B( S d 1 ), set ϵ = f(x) f 0 (x) and observe that any point in K(f 0 ) lies on a unit line segment with centre f 0 (x), for some x S d 1, and that there is a corresponding point f(x) K(f) that lies within ϵ of f 0 (x), since the unit line segment ( is translated ) by no more than ϵ on changing from f 0 to f. Thus ρ K(f), K(f 0 ) ϵ. This argument is symmetric between K(f) and K(f 0 ), and so the desired result follows. We now employ a variant of the arguments of the previous section to prove the density of maps f for which dim H (K(f)) = d, and more strongly for which µ(k(f)) > 0 (where µ(a) denotes the d-dimensional Lebesgue measure of A). Proposition 5.2. The collection of functions f B( S d 1 ) such that is dense. dim H (K(f)) = d 8

9 Proof. Take f 0 B( S d 1 ) and ϵ > 0. Let M > 0 be such that K(f 0 ) is contained in ( M, M) d. Now proceed as in the proof of Lemma 3.2 to find a disjoint family of cubes {Q j } N that cover K(f 0 ), whose sides have length < ϵ/ d with centres a j. Now we do not shift the components of the set K(f 0 ), but rather define an element f B( S d 1 ) by setting f(x) = a j if f 0 (x) Q j ; clearly f f 0 < ϵ. Now we can write K(f) as the finite union K(f) = N U j, where U j consists of all those line segments with centre a j, and observe that N T aj U j = B(0, 1/2). (5.1) Now by the cut-and-move lemma, ( N ) dim H (K(f)) = dim H T aj U j = dim H (B(0, 1/2)) = d. In fact the construction in the above proof gives a little more, namely the density of sets with positive Lebesgue measure. We note that Kakeya sets with zero Lebesgue measure were constructed by Besicovitch [1]. Corollary 5.3. The collection of functions f B( S d 1 ) such that is dense. µ(k(f)) > 0 Proof. We follow the proof above until we achieve (5.1). Then rather than use the shift lemma, we note that since the Lebesgue measure is subadditive N µ(t aj U j ) µ(b(0, 1/2)). It follows that there exists an i such that µ(t ai U i ) µ(b(0, 1/2))/N, and so, since the Lebesgue measure is monotonic and invariant under translations, µ(k(f)) µ(u i ) = µ(t ai U i ) > 0. 9

10 6. A residual collection of Kakeya sets with maximal dimension We now show that a residual collection of Kakeya sets has maximal upper box-counting dimension. Recall that a set is nowhere dense if its closure contains no open sets, and a set is residual if its complement is the countable union of nowhere dense sets, see Oxtoby [8]. We make use of another equivalent definition of the upper box-counting dimension (see Falconer [4], for example). Let N disj (A, ϵ) denote the maximum number of disjoint closed ϵ-balls with centres in A. Then it is well known that log N disj (A, ϵ) dim B (A) = lim sup. (6.1) ϵ 0 log ϵ Theorem 6.1. The collection of functions f B( S d 1 ) such that is a residual subset of B( S d 1 ). Proof. Define dim B (K(f)) = d F m,n = {f B( S d 1 ) : δ < 1/n such that N disj (K(f), δ) > δ 1/m d }. It follows from (6.1) that {f B( S d 1 ) : dim B (K(f)) = d} = {f B( S d 1 ) : dim B (K(f)) = d} = F m,n, n=1 m=1 since dim B (A) = dim B (A). Note that we have already shown that for each m, n N, the set F m,n is dense, since it contains all those f for which dim B (K(f)) = d. Now denote by K (R d ) the collection of all compact subsets of R d, metrised by the symmetric Hausdorff distance dist H. It follows from the result of Lemma 5.1 that the map f K(f) is continuous from B( S d 1 ) into K (R d ). It remains only to show that for any ϵ and r > 0 the set F ϵ,r := {A K (R d ) : δ < ϵ such that N disj (A, δ) > δ r } is open. Given any A F ϵ,r, suppose that there are N > δ r disjoint closed balls of radius δ < ϵ with centres {a j } A. Then these balls are all at 10

11 least some distance η > 0 apart, and so any set B with dist H (A, B) < η/2 contains points b j with a j b j < η/2 and the δ balls with centres {b j } are still disjoint. Thus N disj (B, δ) N > δ r and hence B F ϵ,r, which is sufficient. The result now follows using the Baire Category Theorem, since {f B( S d 1 ) : dim B (K(f)) = d} = m,n F m,n is the countable intersection of open dense sets. An interesting further problem would be to consider whether the Baire generic Kakeya sets (with respect to our parameterisation) also have full Hausdorff (or lower box) dimension and whether they have zero or positive Lebesgue measure. Of course, if one believes the Kakeya conjecture itself, then the Baire generic Kakeya sets also have full Hausdorff dimension, but for the question of measure it is not so clear what to conjecture. Acknowledgments JMF was at the University of Warwick while this work was carried out and was supported by the EPSRC grant EP/J013560/1. JCR was supported by an EPSRC Leadership Fellowship EP/G007470/1. This grant also partially supported the sabbatical visit by EJO to Warwick for the academic year 2013/14 when this work was undertaken. JCR would like to acknowledge the influence of Miles Caddick s Warwick MMath project [2] (supervised by Jose Rodrigo), which provided a comprehensive introduction to the Kakeya Conjecture and its connections with harmonic analysis. The authors are grateful to Alex Iosevich for useful discussions. References [1] Besicovitch, A. (1928) On Kakeya s problem and a similar one. Math. Zeits. 27, [2] Caddick, M. (2014) Multipliers and Bochner Riesz means. Warwick MMath project. [3] Davies, R. (1971) Some remarks on the Kakeya problem. Proc. Cambridge Philos. Soc. 69,

12 [4] Falconer, K. J. (2014) Fractal Geometry: Mathematical Foundations and Applications, John Wiley, 3rd Ed. [5] Katz, N.H. & Tao, T. (2002) New bounds for Kakeya problems. J. Anal. Math. 87, [6] Katz, N.H., Laba, I., & Tao, T. (2000) An improved bound on the Minkowski dimension of Besicovitch sets in R 3. Ann. of Math. 152, [7] Körner, T.W. (2003) Besicovitch via Baire. Studia Math. 158, [8] Oxtoby, J.C. (1980) Measure and category, second edition, Springer- Verlag, New York-Berlin. [9] Robinson, J.C. (2011) Dimensions, Embeddings, and Attractors. Cambridge University Press, Cambridge, UK. [10] Wolff, T. (1995). An improved bound for Kakeya type maximal functions. Rev. Mat. Iberoamericana 11,