KÖRNER’S CONSTRUCTION OF A BESICOVITCH SET

TERENCE TAO

Abstract. We present Tom Körner’s recent argument [6] that (in some sense) “most” Besicovitch sets have measure zero.

1. Introduction

Define a Besicovitch set to be any compact set E ⊆ R^{2} which contains a unit line segment in every direction. A classical result of Besicovitch [2] asserts that such sets can have measure zero. The original construction of Besicovitch is rather complicated, but later authors have given other examples of such sets [1], [3], [4], [6], [7]. Our purpose here is to present the recent argument of Körner which shows in fact that “most” Besicovitch sets (in the sense of the Baire category theorem) have measure zero.

We first give a standard reduction (seen for instance in [3]) that transforms the problem of constructing a Besicovitch set of measure zero, to that of constructing a function from [0, 1] to R whose “projections” - which we will define shortly - also has measure zero.

Definition 1.1. Let f : [0, 1] → R be a measurable function. We define the essential range R(f) ⊆ R of f to be the set of all real numbers y ∈ R such that f^{−1}((y − ε, y + ε)) has positive measure for all ε > 0.

Observe that the essential range is a closed set, and if we change f on a set of measure zero then the essential range is unaffected. Also observe that f is in L^{∞}([0, 1]) if and only if R(f) is bounded. It is easy to show that f(x) ∈ R(f) for almost every x ∈ [0, 1], so in particular by modifying f on a set of measure zero if necessary we may ensure that f(x) ∈ R(f) for all x ∈ [0, 1]. Finally if we perturb f by a bounded function g, then the essential range only changes by ∥g∥_{∞}:

R(f + g) ⊆ R(f) + [−∥g∥_{∞}, ∥g∥_{∞}].  (1)

We define the identity function ι : [0, 1] → R by ι(x) = x, and define the Favard lengths L_{α}(f) for α ∈ R by the formula

L_{α}(f) := |R(f + αι)|;

thus L_{α}(f) is the Lebesgue measure of the range of the function f(x) + αx. Note that if f ∈ L^{∞}([0, 1]), then L_{α}(f) is a finite non-negative number for every α ∈ R.

We now connect this concept to that of a Besicovitch set. Define a quarter-Besicovitch set to be a compact set which contains a unit line segment in every direction from (0, 1) to (1, 1). Clearly one can form a Besicovitch set by pasting together four rotated copies of a quarter-Besicovitch set, so it will suffice to construct quarter-Besicovitch sets of measure zero.

1991 Mathematics Subject Classification. 42A.

Lemma 1.2. Let f ∈ L^{∞}([0, 1]) be a function such that L_{α}(f) = 0 for all α ∈ [0, 1]. Then the set

E := {(x, y) : x ∈ R(f + yι), y ∈ [0, 1]}

is a Besicovitch set of measure zero. In fact, the intersection of E with every horizontal line {(x, y) : x ∈ R} also has one-dimensional measure zero.

Proof By construction it is clear that the intersection of E with every horizontal line has measure zero, thus by Fubini’s theorem E also has measure zero. E is also clearly bounded. To show that E is closed, let (x_{n}, y_{n}) be a sequence in E which converges to (x, y). Observe from (1) that

x_{n} ∈ R(f + y_{n}ι) ⊆ R(f + yι) + [−|y − y_{n}|, |y − y_{n}|].

Taking limits and noting that R(f + yι) is closed, we obtain x ∈ R(f + yι), thus (x, y) ∈ E and so E is closed, hence compact.

It remains to show that E is a quarter-Besicovitch set. If it were true that f(x) + yx ∈ R(f + yι) for all x ∈ [0, 1] and y ∈ [0, 1], then this would be immediate since the line segment {(f(y) + yx, x) : x ∈ [0, 1]} would then have length at least 1, be oriented in the direction (y, 1), and be contained in E for every y ∈ [0, 1].

However, the definition of essential range does not quite let us ensure that f(x)+ yx ∈ R(f + yι) for all x and y; for any fixed y, we can only ensure this for almost every x. But then we can modify f on a set of measure zero (which does not affect E) so that f(x) + yx ∈ R(f + yι) for all rational y ∈ [0, 1]. We can then pass from rational y to real y by using the fact, derived above, that E is closed. The preceding argument then allows us to conclude that E is a quarter-Besicovitch set.

It remains to show that there exists functions f for which L_{α}(f) vanishes identically. We begin with some basic observations about L_{α}. Since L_{α}(f) = L_{0}(f + αι), we can begin by discussing L_{0}.

Lemma 1.3. The function L_{0} : L^{∞}([0, 1]) → R^{+} is upper semi-continuous; in other words, for every f ∈ L^{∞}([0, 1]) and every ε > 0 there exists an r > 0 such that L_{0}(f + g) ≤ L_{0}(f) + ε whenever ∥g∥_{∞} ≤ r.

Proof Since R(f) is closed, we have

R(f) = ⋂_{n>0} R(f) + [−\frac{1}{n}, \frac{1}{n}]

and hence there exists an r > 0 such that

|R(f) + [−r, r]| ≤ |R(f)| + ε.

The claim then follows from (1).

We can in fact make the upper semi-continuity uniform on compacta in α:

Corollary 1.4. Let K be a compact subset of R. Then for every f ∈ L^{∞}([0, 1]) and every ε > 0 there exists an r > 0 such that L_{α}(f + g) ≤ L_{α}(f) + ε whenever α ∈ K and ∥g∥_{∞} ≤ r.

Proof For any fixed α = α_{0} the claim is true from Lemma 1.3, and in fact one can make the claim true for all α in an open neighborhood of α_{0} (since any perturbation of the α parameter can be absorbed into the g perturbation). The claim then follows by compactness of K.

Our objective is to make L_{α}(f) vanish for all α. We begin by making L_{0}(f) vanish. Define a piecewise constant function f ∈ L^{∞}([0, 1]) to be any finite linear combination of characteristic functions χ_{I} of intervals I ⊆ [0, 1], and define P C to be the space of all piecewise constant functions. Clearly we have L_{0}(f) = 0 for all f ∈ P C, since the essential range of a piecewise constant function is finite. More generally, we have

Lemma 1.5. If f ∈ P C, then L_{α}(f) ≤ |α| for all α ∈ R.

Proof Let f = ∑_{I} c_{I}χ_{I}, where the c_{I} are real numbers and I ranges over a finite partition of [0, 1] into intervals. Then R(f + αι) = ⋃_{I}(c_{I} + αI), and hence

L_{α}(f) = | ⋃_{I}(c_{I} + αI)| ≤ ∑_{I} |α||I| = |α|

as desired.

Now define \overline{P C} to be the closure^{1} of P C in L^{∞}([0, 1]). This is clearly a Banach space with the L^{∞} metric. It clearly contains the identity function ι (since we can approximate ι by a piecewise constant “staircase function”).

We can now prove the following result (which is essentially in [6])

Theorem 1.6. The set

\{f ∈ \overline{P C} : L_{α}(f) > 0 for some α ∈ [0, 1]\}  (2)

is of the first category in \overline{P C} (i.e. it is the countable union of nowhere dense sets in \overline{P C}. In particular, by the Baire category theorem the set

\{f ∈ \overline{P C} : L_{α}(f) = 0 for all α ∈ [0, 1]\}  (3)

is of the second category, and hence non-empty.

Combining this result with Lemma 1.2 we obtain a large class of Besicovitch sets. Proof It suffices to show that the sets

\{f ∈ \overline{P C} : L_{α}(f) ≥ 2ε for some α ∈ [y, y + ε]\}  (4)

is nowhere dense for all 0 < ε ≤ 1 and y ∈ [0, 1], since one can clearly contain (2) in a countable union of sets of the form (4).

Fix y, ε. By translating f ↦ f − yι if necessary we may assume that y = 0 (note that this map is of course a homeomorphism on \overline{P C}). By Corollary 1.4, the set (4) is closed in \overline{P C}. But from Lemma 1.5, the set (4) is disjoint from P C, which is of course dense in \overline{P C}. Hence the set (4) is nowhere dense, as desired.

Observe that the above proof, combined with the standard proof of the Baire category theorem, in fact gives a fairly constructive way to obtain an explicit function f in the set (3). Given any f ∈ \overline{P C}, and any α ∈ [0, 1], we may perturb f by an arbitrarily small amount in L^{∞} norm to ensure that f + αι lies in P C, which will make L_{α′}(f) ≤ |α − α′| for all α′ ∈ R. If we then enumerate the rationals in [0, 1] as α_{1}, α_{2}, … and then perform these small perturbations on each rational in turn, and then pass to the limit in the complete metric space \overline{P C}, we can obtain a function f with L_{α}(f) = 0 for all α ∈ [0, 1] (indeed one can even ensure that L_{α}(f) = 0 for all α ∈ R). Note that we need the uniform upper semi-continuity property from Corollary 1.4 to ensure that later perturbations to not totally undo the efforts of earlier perturbations to make L_{α}(f) small, providing of course that the perturbations are always chosen sufficiently small.

1This space is not all of L^{∞}([0, 1]) - for instance, sin( \frac{1}{x}) is not in \overline{P C}; however the space contains all functions of bounded variation, and more generally of bounded q-variation for any 1 ≤ q < ∞. The space \overline{P C} is in fact some sort of limiting version of the q-variation spaces V_{q} as q → ∞, in much the way that the sequence space c_{0} is sort of a limiting version of l^{q} as q → ∞.

One could even modify this argument to give a discretized version, where for any δ > 0 we can construct a set containing a δ × 1 rectangle in every direction, and with total area at most c(δ), where c(δ) → 0 as δ → 0. However the actual value of c(δ) obtained will be very weak, mostly due to the poor dependence of r on ε in Lemma 1.3, which becomes very bad if the complexity of R(f) (i.e. the number of connected components of R(f)) deteriorates). Indeed, the optimal value of c(δ) is O(1/ log 1/δ); see [4], whereas a naive implementation of the argument here seems to give something closer to c(δ) = O(1/ log_{∗} 1/δ), where log_{∗} is the inverse of the tower function T (n), defined by T (1) = 1 and T (n + 1) = 2^{T (n)}. Interestingly, this inverse tower function has appeared in a closely related context [5].

References

[1] R. Alexander, Random compact sets related to the Kakeya problem, Proc. Amer. Math. Soc. 53 (1975), 415–419.

[2] A. S. Besicovitch, On Kakeya’s problem and a similar one, Mat. Zeit. 27 (1928), 312–320.

[3] J.-P. Kahane, Trois notes sur les ensembles parfaits linéaires, Enseignment Math. 15 (1969), 185–192.

[4] U. Keich, On L^{p} bounds for Kakeya maximal functions and the Minkowski dimension in R^{2}, Bull. London Math. Soc. 31 (1999), 213–221.

[5] Y. Peres, B. Solomyak, How likely is Buffon’s needle to fall near a planar Cantor set?, Pac. J. Math. 204 (2002), 473–496.

[6] T. Körner, Besicovitch via Baire, preprint.

[7] T. Wolff, Recent work connected with the Kakeya problem, Prospects in mathematics (Princeton, NJ, 1996), 129–162, Amer. Math. Soc., Providence, RI, 1999.

Department of Mathematics, UCLA, Los Angeles, CA 90024

   E-mail address: tao@math.ucla.edu