If a compact set K \subset R^2 contains a positive-dimensional family of
line-segments in positively many directions, then K has positive measure.Comment: 7 pages. v2: corrected typos and added references, to appear in Publ.
  Ma

Orponen, Tuomas

English

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If a compact set K ⊂ ℝ2 contains a positive-dimensional family of line-segments in every direction, then K has positive measure

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Publ. Mat. 58 (2014), 213–220DOI: 10.5565/PUBLMAT 58114 10NON-EXISTENCE OF MULTI-LINE BESICOVITCHSETSTuomas OrponenAbstract: If a compact set K ⊂ R2 contains a positive-dimensional family of line-segments in every direction, then K has positive measure.2010 Mathematics Subject Classification: Primary: 28A80; Secondary: 28A78,42B25.Key words: Besicovitch sets, Kakeya maximal operator.1. IntroductionThere are compact sets in the plane, which have zero Lebesgue mea-sure, yet contain a line segment in every direction. Such a set was firstconstructed by A. S. Besicovitch in 1919, and this existence result is nowone of the most widely known theorems in geometric measure theory, notleast due to its profound consequences for Euclidean harmonic analysis.For a historical account of the problem and (some of) its connections,see [Fa, §7]. In the present paper, we ask: what if a compact set con-tains many lines in every direction – or even many directions? If theword ‘many’ is interpreted as in Theorems 1.1 and 1.2, the conclusionis that the set has to have positive Lebesgue measure. In other words,there exist no ‘multi-line’ Besicovitch sets. Our proof uses methods inharmonic analysis. More precisely, we extend Co´rdoba’s proof [Co] forthe ‘almost boundedness’ of the Kakeya maximal operator from 1977.Given a direction e ∈ S1 and a number s ∈ [0, 1], a family of line-segments L perpendicular to e is called s-dimensional, if the union L =∪L ⊂ R2 satisfiesHs(ρe(L)) > 0,where ρe stands for the orthogonal projection ρe(x) = x ·e, and Hs is thes-dimensional Hausdorff measure, see [Ma, Chapter 4]. In case s > 0, thecollection L is positive-dimensional. The definition imposes no conditionsThe author gratefully acknowledges the financial support of the Finnish NationalDoctoral Programme in Mathematics and its Applications.214 T. Orponenof measurability on L or ρe(L), even though we will actually need to knowthat the projections ρe(L) are regular enough for Frostman’s lemma tobe applied. Fortunately, this is automatically satisfied, see Lemma 4.4.Our main result is:Theorem 1.1. Let K ⊂ R2 be a compact set containing the unionsof positive-dimensional families of line-segments in H1-positively manydirections. Then L2(K) > 0.It is not assumed that the set with ‘H1-positively many directions’ ismeasurable. Theorem 1.1 is a corollary of the following slightly sharperresult:Theorem 1.2. Let 0 < s ≤ 1, and let K ⊂ R2 be a compact set con-taining the unions of s-dimensional families of line-segments in a set ofdirections E0 ⊂ S1 with dimE0 > 1− s. Then L2(K) > 0.Again, we require no regularity from E. A word on notation beforewe begin: we write A . B, if there exists a finite constant C > 0 suchthat A ≤ CB. If C is allowed to depend on a parameter, say, p, we maywrite A .p B. The sequence A . B . A is abbreviated to A  B.Added in proof: After the paper was accepted, the author was informedthat I.  Laba and T. Tao have previously obtained a result similar toTheorem 1.1, see [LT, Corollary 1.3].2. AcknowledgementsI am grateful to an anonymous referee for a careful reading of themanuscript and several useful comments.3. The multi-line maximal operatorAdopting Co´rdoba’s approach in proving that ordinary Kakeya setshave dimension two, we first need to introduce a maximal operator suit-able for our purposes. Our operator, defined on S1 rather than R2, issimilar to the modification of Co´rdoba’s operator introduced by J. Bour-gain in [Bo]. If e ∈ S1 and δ > 0, we denote by T δe the collection ofdisjoint δ-tubes perpendicular to the line spanned by e. More precisely,if ρe : R2 → R is the orthogonal projection ρe(x) = x · e, we setT δe := {ρ−1e [jδ, (j + 1)δ) : j ∈ Z}.Next, we introduce the family of (δ, e)-rectangles, denoted by Rδe. Arectangle R ⊂ R2 is a member of Rδe, if R is a δ×1-rectangle, and R ⊂ TNon-Existence of Multi-Line Besicovitch Sets 215for some T ∈ T δe . A set B ⊂ R2 is called a (δ, e)-set, if B ∩ T ∈ Rδe forevery tube T ∈ T δe . Given 0 < s ≤ 1, a measure µ on R2 is called a(δ, e, s)-measure, if µ is actually a function with the following properties:(i) there exists a (δ, e)-set B ⊂ R2 such thatµ =∑T∈T δeaT · χB∩T ,(ii) the L1-norm of µ is bounded by one,δ∑T∈T δeaT = ‖µ‖L1(R2) ≤ 1,(iii) the projection µe := ρe]µ satisfies the growth conditionµe(I) ≤ `(I)sfor every interval I ⊂ R.The parameter s > 0 will be thought as fixed, and the collection of all(δ, e, s)-measures is simply denoted by Mδe.Definition 3.1 (Multi-line maximal operator). If f : R2 → R is a bound-ed Borel function, we setMδf(e) := sup{∫f dµ : µ ∈Mδe}.Remark 3.2. For a fixed bounded Borel function f : R2 → R, the map-ping e 7→ Mδf(e) is lower semicontinuous. Indeed, if µe ∈ Mδe, wemay rotate µe to obtain measures µξ ∈ Mδξ, for ξ 6= e. As ξ → e, the,difference∫f dµe−∫ f dµξ tends to zero. As a consequence of the semi-continuity, the function M δf can be discretized by choosing a finite col-lection of vectors {e1, . . . , eq} ⊂ S1, corresponding measures µej ∈ Mδejand some numbers δj < δ in such a manner that S1 is covered by theballs B(ej , δj), andMδf(e) q∑j=1[∫f dµejχB(ej ,δj)(e)].In the following proofs, all measurability issues can be resolved by re-placing Mδf with the discretized version.216 T. Orponen4. A restricted weak-type (2,2) bound for MδFix s ∈ (0, 1) for the rest of the paper. The central component in theproof of Theorem 1.2 is the following estimate:Proposition 4.1. Let σ be a Borel measure on S1 satisfying the boundσ(B(e, r)) ≤ r1−t, e ∈ S1, r > 0, for some t < s. Then, the maximaloperator M δ satisfies the weak-type (2, 2)-estimate(4.2) σ({e ∈ S1 : M δχB(e) ≥ λ})1/2 .t L2(B)1/2λ, λ > 0,for compact sets B ⊂ R2, where the implicit constants are independentof δ > 0.Proof: Fix λ > 0 and write E := {e ∈ S1 : MδsχB(e) ≥ λ}. Foreach e ∈ E, choose a measure µe ∈ Mδe with∫Bdµe & λ. Then, sincethe measures µe are functions, we may estimate as follows:λσ(E) .∫E∫Bdµe dσ(e) =∫B∫Eµe(x) dσ(e) dx≤ L2(B)1/2(∫ (∫Eµe(x) dσ(e))2dx)1/2= L2(B)1/2(∫∫E×E[∫µe(x)µξ(x) dx]dσ(e) dσ(ξ))1/2.(4.3)So, it remains to bound the correlation∫µe(x)µξ(x) dx.We first write ∫µe(x)µξ(x) dx =∑j∈Zaξj∫Rξjµe(x) dx,where Rξj ∈ Rδξ is some (δ, ξ)-rectangle, on which µξ takes the constantvalue aξj , see Figure 1.Non-Existence of Multi-Line Besicovitch Sets 217T ξjFigure 1. The tube T ξj ⊃Rξj intersecting the rectangles Rei .Similarly expanding µe as a sum µe =∑aeiχRei , we have∫Rξjµe(x) dx ≤∑i:Rei∩Rξj 6=∅aei · L2(Rei ∩Rξj), j ∈ Z.The diameter of the intersection Rei ∩Rξj is bounded bydiam(Rei ∩Rξj) .δ|e− ξ|+ δ ,so L2(Rei ∩ Rξj) . δ2/(|e − ξ| + δ). Finally, we have to estimate thesum of the numbers aei over the indices {i : Rei ∩ Rξj 6= ∅}. Using basictrigonometry, the projection of the rectangle Rξj onto the line spannedby e is an interval Ie,ξ of length`(Ie,ξ) . |e− ξ|+ δ.In particular, the rectangles Rei with Rei ∩ Rξj 6= ∅ are all contained inthe pre-image ρ−1e (2Ie,ξ). Combining this with assumption (iii),∑i:Rei∩Rξjaei =δ−1 ∑i:Rei∩Rξj 6=∅∫Reiµe(x) dx≤δ−1(µe)e(2Ie,ξ) . (|e− ξ|+ δ)sδ.218 T. OrponenPutting everything together and using assumption (ii) yields∫µe(x)µξ(x) dx .∑j∈Z(aξj ·δ2|e− ξ|+ δ ·(|e− ξ|+ δ)sδ)≤ |e− ξ|s−1∑j∈Zδ · aξj ≤ |e− ξ|s−1.Inserting this back into (4.3) leads toλσ(E).L2(B)1/2(∫E[∫S1|e− ξ|s−1 σ(ξ)]dσ(e))1/2.L2(B)1/2σ(E)1/2.This concludes the proof of (4.2). The growth bound assumed from σwas used above to obtain∫S1|e− ξ|s−1 dσ(e) .t 1.The proof of this is standard issue, see for example [Ma, p. 109].The following lemma addresses the measurability issues related to theprojections ρe(L), mentioned at the beginning of the introduction.Lemma 4.4. Let K ⊂ R2 be a compact set, let e ∈ S1 and let c > 0.Let L be the collection of all line-segments contained in K, which areperpendicular to e and have length at least c. Then L := ∪L is compact;in particular, ρe(L) is compact.Proof: Of course, we only need to verify that L is closed. Fix x ∈ L.We first pick a sequence of points (xi)i∈N in L with xi → x, and notethat each point xi is contained in some line-segment `i ∈ L. We thenuse the Blaschke selection theorem [Fa, Theorem 3.16] to produce asubsequence (`ij )j∈N, convergent in the Hausdorff metric to a compactset ` ⊂ K. It is clear from the definition of convergence in the Hausdorffmetric that x ∈ `, and ` is a subset of the line ρ−1e {t}, where t := ρe(x).Moreover, ` is connected according to [Fa, Theorem 3.18]. Thus, ` iseither a line-segment or a point contained in K ∩ ρ−1e {t}. Finally, weobserve that ` has length at least c, since the δ-neighborhoods of ` containsome line-segments `i for any δ > 0. We conclude that ` ∈ L, and sox ∈ ` ⊂ L.Now we are equipped to prove Theorem 1.2Proof of Theorem 1.2: Let K ⊂ R2 be as in Theorem 1.2. Let Lec be thecollection of all line-segments contained in K, perpendicular to e andwith length at least c > 0. Write Le := ∪Lec. Choosing c > 0 and α > 0Non-Existence of Multi-Line Besicovitch Sets 219small enough, the set E = {e ∈ S1 : Hs(ρe(Le)) ≥ α} has dimensiondimE > s − 1, where Hs stands for s-dimensional Hausdorff content.Without loss of generality, we may assume that c = 1. Thus, Le := Le1consists of line-segments of length at least one, and ρe(Le) is a compactset according to Lemma 4.4. We can now apply Frostman’s lemma tothe sets ρe(Le): it follows from the standard proof of this lemma, see[Ma, Theorem 8.8], that for every e ∈ E we can locate a measure ν˜e,supported on ρe(Le), such that ν˜e(I) ≤ `(I)s for every interval I ⊂ R,and 1 . ν˜e(R) ≤ 1, where the implicit constants only depend on α and s.To produce from ν˜e a measure in Mδe, we first discretize ν˜e by definingνe =110∑j∈Zν˜e[jδ, (j + 1)δ)δχ[jδ,(j+1)δ).Then νe is a measure with total mass νe(R) = ν˜e(R)/10 & 1. Thefactor 1/10 is there only to ensure that νe satisfies the growth conditionνe(I) ≤ `(I)s. Whenever νe[jδ, (j + 1)δ) = ν˜e[jδ, (j + 1)δ) > 0, weknow that the intersection spt ν˜e ∩ [jδ, (j + 1)δ] is non-empty. Recallingthe definition of ν˜e, this means that the intersection ρ−1e [jδ, (j + 1)δ] ∩K contains an entire unit line-segment. It follows that we may find arectangle Rj ∈ Rδe contained in the intersection ρ−1e [jδ, (j+ 1)δ)∩K(δ).For each j, we choose one – and only one – such rectangle Rj and defineµe =∑j∈Zνe[jδ, (j + 1)δ)δχRj .Then the projection (µe)e coincides with νe and, consequently, satisfiesthe growth condition in assumption (iii); the assumptions (i) and (ii) areclearly satisfied as well. We conclude that µe ∈Mδe, whenceMδχK(δ)(e) ≥∫K(δ)dµe = µe(R2) & 1.This holds for every direction e ∈ E, so there exists a constant m > 0such that the setsEδ := {e ∈ S1 : MδχK(δ)(e) > m}contain E, for every δ > 0. In particular, we may find t < s suchthat the numbers H1−t(Eδ) have a uniform lower bound β > 0. Thesets Eδ are open, so another application of (the proof of) Frostman’slemma yields measures σδ, δ > 0, supported on Eδ and satisfying thebounds σδ(Eδ) & 1 and σδ(B(e, r)) ≤ r1−t, where the implicit constantsare again independent of δ > 0. It remains to apply the weak-type220 T. Orponenestimate (4.2) as follows:1 . σδ(Eδ) = σδ({e ∈ S1 : MδχK(δ)(e) > m}) . [L2(K(δ))]2c2.Letting δ → 0 shows that L2(K) > 0 and completes the proof of Theo-rem 1.2.References[Bo] J. Bourgain, Besicovitch type maximal operators and applica-tions to Fourier analysis, Geom. Funct. Anal. 1(2) (1991), 147–187.DOI: 10.1007/BF01896376.[Co] A. Cordoba, The Kakeya maximal function and the sphericalsummation multipliers, Amer. J. Math. 99(1) (1977), 1–22.[Fa] K. J. Falconer, “The geometry of fractal sets”, CambridgeTracts in Mathematics 85, Cambridge University Press, Cam-bridge, 1986.[LT] I.  Laba and T. Tao, An x-ray transform estimate in Rn, Rev.Mat. Iberoamericana 17(2) (2001), 375–407. DOI: 10.4171/RMI/298.[Ma] P. Mattila, “Geometry of sets and measures in Euclideanspaces”, Fractals and rectifiability, Cambridge Studies in AdvancedMathematics 44, Cambridge University Press, Cambridge, 1995.Department of Mathematics and StatisticsUniversity of HelsinkiFI-00014 HelsinkiFinlandE-mail address: tuomas.orponen@helsinki.fiPrimera versio´ rebuda el 16 de novembre de 2012,darrera versio´ rebuda el 31 de gener de 2013.

Non-existence of multi-line Besicovitch sets

T. Orponen

Crossref

Publicacions Matemàtiques

Revistes Catalanes amb Accés Obert

Publ. Mat. 58 (2014), 213–220
DOI: 10.5565/PUBLMAT 58114 10
NON-EXISTENCE OF MULTI-LINE BESICOVITCH
SETS
Tuomas Orponen
Abstract: If a compact set K ⊂ R2 contains a positive-dimensional family of line-
segments in every direction, then K has positive measure.
2010 Mathematics Subject Classification: Primary: 28A80; Secondary: 28A78,
42B25.
Key words: Besicovitch sets, Kakeya maximal operator.
1. Introduction
There are compact sets in the plane, which have zero Lebesgue mea-
sure, yet contain a line segment in every direction. Such a set was first
constructed by A. S. Besicovitch in 1919, and this existence result is now
one of the most widely known theorems in geometric measure theory, not
least due to its profound consequences for Euclidean harmonic analysis.
For a historical account of the problem and (some of) its connections,
see [Fa, §7]. In the present paper, we ask: what if a compact set con-
tains many lines in every direction – or even many directions? If the
word ‘many’ is interpreted as in Theorems 1.1 and 1.2, the conclusion
is that the set has to have positive Lebesgue measure. In other words,
there exist no ‘multi-line’ Besicovitch sets. Our proof uses methods in
harmonic analysis. More precisely, we extend Co´rdoba’s proof [Co] for
the ‘almost boundedness’ of the Kakeya maximal operator from 1977.
Given a direction e ∈ S1 and a number s ∈ [0, 1], a family of line-
segments L perpendicular to e is called s-dimensional, if the union L =
∪L ⊂ R2 satisfies
Hs(ρe(L)) > 0,
where ρe stands for the orthogonal projection ρe(x) = x ·e, and Hs is the
s-dimensional Hausdorff measure, see [Ma, Chapter 4]. In case s > 0, the
collection L is positive-dimensional. The definition imposes no conditions
The author gratefully acknowledges the financial support of the Finnish National
Doctoral Programme in Mathematics and its Applications.
214 T. Orponen
of measurability on L or ρe(L), even though we will actually need to know
that the projections ρe(L) are regular enough for Frostman’s lemma to
be applied. Fortunately, this is automatically satisfied, see Lemma 4.4.
Our main result is:
Theorem 1.1. Let K ⊂ R2 be a compact set containing the unions
of positive-dimensional families of line-segments in H1-positively many
directions. Then L2(K) > 0.
It is not assumed that the set with ‘H1-positively many directions’ is
measurable. Theorem 1.1 is a corollary of the following slightly sharper
result:
Theorem 1.2. Let 0 < s ≤ 1, and let K ⊂ R2 be a compact set con-
taining the unions of s-dimensional families of line-segments in a set of
directions E0 ⊂ S1 with dimE0 > 1− s. Then L2(K) > 0.
Again, we require no regularity from E. A word on notation before
we begin: we write A . B, if there exists a finite constant C > 0 such
that A ≤ CB. If C is allowed to depend on a parameter, say, p, we may
write A .p B. The sequence A . B . A is abbreviated to A  B.
Added in proof: After the paper was accepted, the author was informed
that I.  Laba and T. Tao have previously obtained a result similar to
Theorem 1.1, see [LT, Corollary 1.3].
2. Acknowledgements
I am grateful to an anonymous referee for a careful reading of the
manuscript and several useful comments.
3. The multi-line maximal operator
Adopting Co´rdoba’s approach in proving that ordinary Kakeya sets
have dimension two, we first need to introduce a maximal operator suit-
able for our purposes. Our operator, defined on S1 rather than R2, is
similar to the modification of Co´rdoba’s operator introduced by J. Bour-
gain in [Bo]. If e ∈ S1 and δ > 0, we denote by T δe the collection of
disjoint δ-tubes perpendicular to the line spanned by e. More precisely,
if ρe : R2 → R is the orthogonal projection ρe(x) = x · e, we set
T δe := {ρ−1e [jδ, (j + 1)δ) : j ∈ Z}.
Next, we introduce the family of (δ, e)-rectangles, denoted by Rδe. A
rectangle R ⊂ R2 is a member of Rδe, if R is a δ×1-rectangle, and R ⊂ T
Non-Existence of Multi-Line Besicovitch Sets 215
for some T ∈ T δe . A set B ⊂ R2 is called a (δ, e)-set, if B ∩ T ∈ Rδe for
every tube T ∈ T δe . Given 0 < s ≤ 1, a measure µ on R2 is called a
(δ, e, s)-measure, if µ is actually a function with the following properties:
(i) there exists a (δ, e)-set B ⊂ R2 such that
µ =
∑
T∈T δe
aT · χB∩T ,
(ii) the L1-norm of µ is bounded by one,
δ
∑
T∈T δe
aT = ‖µ‖L1(R2) ≤ 1,
(iii) the projection µe := ρe]µ satisfies the growth condition
µe(I) ≤ `(I)s
for every interval I ⊂ R.
The parameter s > 0 will be thought as fixed, and the collection of all
(δ, e, s)-measures is simply denoted by Mδe.
Definition 3.1 (Multi-line maximal operator). If f : R2 → R is a bound-
ed Borel function, we set
Mδf(e) := sup
{∫
f dµ : µ ∈Mδe
}
.
Remark 3.2. For a fixed bounded Borel function f : R2 → R, the map-
ping e 7→ Mδf(e) is lower semicontinuous. Indeed, if µe ∈ Mδe, we
may rotate µe to obtain measures µξ ∈ Mδξ, for ξ 6= e. As ξ → e, the,
difference
∫
f dµe−∫ f dµξ tends to zero. As a consequence of the semi-
continuity, the function M δf can be discretized by choosing a finite col-
lection of vectors {e1, . . . , eq} ⊂ S1, corresponding measures µej ∈ Mδej
and some numbers δj < δ in such a manner that S
1 is covered by the
balls B(ej , δj), and
Mδf(e) 
q∑
j=1
[∫
f dµejχB(ej ,δj)(e)
]
.
In the following proofs, all measurability issues can be resolved by re-
placing Mδf with the discretized version.
216 T. Orponen
4. A restricted weak-type (2,2) bound for Mδ
Fix s ∈ (0, 1) for the rest of the paper. The central component in the
proof of Theorem 1.2 is the following estimate:
Proposition 4.1. Let σ be a Borel measure on S1 satisfying the bound
σ(B(e, r)) ≤ r1−t, e ∈ S1, r > 0, for some t < s. Then, the maximal
operator M δ satisfies the weak-type (2, 2)-estimate
(4.2) σ({e ∈ S1 : M δχB(e) ≥ λ})1/2 .t L
2(B)1/2
λ
, λ > 0,
for compact sets B ⊂ R2, where the implicit constants are independent
of δ > 0.
Proof: Fix λ > 0 and write E := {e ∈ S1 : MδsχB(e) ≥ λ}. For
each e ∈ E, choose a measure µe ∈ Mδe with
∫
B
dµe & λ. Then, since
the measures µe are functions, we may estimate as follows:
λσ(E) .
∫
E
∫
B
dµe dσ(e) =
∫
B
∫
E
µe(x) dσ(e) dx
≤ L2(B)1/2
(∫ (∫
E
µe(x) dσ(e)
)2
dx
)1/2
= L2(B)1/2
(∫∫
E×E
[∫
µe(x)µξ(x) dx
]
dσ(e) dσ(ξ)
)1/2
.
(4.3)
So, it remains to bound the correlation∫
µe(x)µξ(x) dx.
We first write ∫
µe(x)µξ(x) dx =
∑
j∈Z
aξj
∫
Rξj
µe(x) dx,
where Rξj ∈ Rδξ is some (δ, ξ)-rectangle, on which µξ takes the constant
value aξj , see Figure 1.
Non-Existence of Multi-Line Besicovitch Sets 217
T ξj
Figure 1. The tube T ξj ⊃Rξj intersecting the rectangles Rei .
Similarly expanding µe as a sum µe =
∑
aeiχRei , we have∫
Rξj
µe(x) dx ≤
∑
i:Rei∩Rξj 6=∅
aei · L2(Rei ∩Rξj), j ∈ Z.
The diameter of the intersection Rei ∩Rξj is bounded by
diam(Rei ∩Rξj) .
δ
|e− ξ|+ δ ,
so L2(Rei ∩ Rξj) . δ2/(|e − ξ| + δ). Finally, we have to estimate the
sum of the numbers aei over the indices {i : Rei ∩ Rξj 6= ∅}. Using basic
trigonometry, the projection of the rectangle Rξj onto the line spanned
by e is an interval Ie,ξ of length
`(Ie,ξ) . |e− ξ|+ δ.
In particular, the rectangles Rei with R
e
i ∩ Rξj 6= ∅ are all contained in
the pre-image ρ−1e (2Ie,ξ). Combining this with assumption (iii),∑
i:Rei∩Rξj
aei =δ
−1 ∑
i:Rei∩Rξj 6=∅
∫
Rei
µe(x) dx≤δ−1(µe)e(2Ie,ξ) . (|e− ξ|+ δ)
s
δ
.
218 T. Orponen
Putting everything together and using assumption (ii) yields∫
µe(x)µξ(x) dx .
∑
j∈Z
(
aξj ·
δ2
|e− ξ|+ δ ·
(|e− ξ|+ δ)s
δ
)
≤ |e− ξ|s−1
∑
j∈Z
δ · aξj ≤ |e− ξ|s−1.
Inserting this back into (4.3) leads to
λσ(E).L2(B)1/2
(∫
E
[∫
S1
|e− ξ|s−1 σ(ξ)
]
dσ(e)
)1/2
.L2(B)1/2σ(E)1/2.
This concludes the proof of (4.2). The growth bound assumed from σ
was used above to obtain∫
S1
|e− ξ|s−1 dσ(e) .t 1.
The proof of this is standard issue, see for example [Ma, p. 109].
The following lemma addresses the measurability issues related to the
projections ρe(L), mentioned at the beginning of the introduction.
Lemma 4.4. Let K ⊂ R2 be a compact set, let e ∈ S1 and let c > 0.
Let L be the collection of all line-segments contained in K, which are
perpendicular to e and have length at least c. Then L := ∪L is compact;
in particular, ρe(L) is compact.
Proof: Of course, we only need to verify that L is closed. Fix x ∈ L.
We first pick a sequence of points (xi)i∈N in L with xi → x, and note
that each point xi is contained in some line-segment `i ∈ L. We then
use the Blaschke selection theorem [Fa, Theorem 3.16] to produce a
subsequence (`ij )j∈N, convergent in the Hausdorff metric to a compact
set ` ⊂ K. It is clear from the definition of convergence in the Hausdorff
metric that x ∈ `, and ` is a subset of the line ρ−1e {t}, where t := ρe(x).
Moreover, ` is connected according to [Fa, Theorem 3.18]. Thus, ` is
either a line-segment or a point contained in K ∩ ρ−1e {t}. Finally, we
observe that ` has length at least c, since the δ-neighborhoods of ` contain
some line-segments `i for any δ > 0. We conclude that ` ∈ L, and so
x ∈ ` ⊂ L.
Now we are equipped to prove Theorem 1.2
Proof of Theorem 1.2: Let K ⊂ R2 be as in Theorem 1.2. Let Lec be the
collection of all line-segments contained in K, perpendicular to e and
with length at least c > 0. Write Le := ∪Lec. Choosing c > 0 and α > 0
Non-Existence of Multi-Line Besicovitch Sets 219
small enough, the set E = {e ∈ S1 : Hs(ρe(Le)) ≥ α} has dimension
dimE > s − 1, where Hs stands for s-dimensional Hausdorff content.
Without loss of generality, we may assume that c = 1. Thus, Le := Le1
consists of line-segments of length at least one, and ρe(L
e) is a compact
set according to Lemma 4.4. We can now apply Frostman’s lemma to
the sets ρe(L
e): it follows from the standard proof of this lemma, see
[Ma, Theorem 8.8], that for every e ∈ E we can locate a measure ν˜e,
supported on ρe(Le), such that ν˜e(I) ≤ `(I)s for every interval I ⊂ R,
and 1 . ν˜e(R) ≤ 1, where the implicit constants only depend on α and s.
To produce from ν˜e a measure in Mδe, we first discretize ν˜e by defining
νe =
1
10
∑
j∈Z
ν˜e[jδ, (j + 1)δ)
δ
χ[jδ,(j+1)δ).
Then νe is a measure with total mass νe(R) = ν˜e(R)/10 & 1. The
factor 1/10 is there only to ensure that νe satisfies the growth condition
νe(I) ≤ `(I)s. Whenever νe[jδ, (j + 1)δ) = ν˜e[jδ, (j + 1)δ) > 0, we
know that the intersection spt ν˜e ∩ [jδ, (j + 1)δ] is non-empty. Recalling
the definition of ν˜e, this means that the intersection ρ−1e [jδ, (j + 1)δ] ∩
K contains an entire unit line-segment. It follows that we may find a
rectangle Rj ∈ Rδe contained in the intersection ρ−1e [jδ, (j+ 1)δ)∩K(δ).
For each j, we choose one – and only one – such rectangle Rj and define
µe =
∑
j∈Z
νe[jδ, (j + 1)δ)
δ
χRj .
Then the projection (µe)e coincides with ν
e and, consequently, satisfies
the growth condition in assumption (iii); the assumptions (i) and (ii) are
clearly satisfied as well. We conclude that µe ∈Mδe, whence
MδχK(δ)(e) ≥
∫
K(δ)
dµe = µe(R2) & 1.
This holds for every direction e ∈ E, so there exists a constant m > 0
such that the sets
Eδ := {e ∈ S1 : MδχK(δ)(e) > m}
contain E, for every δ > 0. In particular, we may find t < s such
that the numbers H1−t(Eδ) have a uniform lower bound β > 0. The
sets Eδ are open, so another application of (the proof of) Frostman’s
lemma yields measures σδ, δ > 0, supported on Eδ and satisfying the
bounds σδ(Eδ) & 1 and σδ(B(e, r)) ≤ r1−t, where the implicit constants
are again independent of δ > 0. It remains to apply the weak-type
220 T. Orponen
estimate (4.2) as follows:
1 . σδ(Eδ) = σδ({e ∈ S1 : MδχK(δ)(e) > m}) . [L
2(K(δ))]2
c2
.
Letting δ → 0 shows that L2(K) > 0 and completes the proof of Theo-
rem 1.2.
References
[Bo] J. Bourgain, Besicovitch type maximal operators and applica-
tions to Fourier analysis, Geom. Funct. Anal. 1(2) (1991), 147–187.
DOI: 10.1007/BF01896376.
[Co] A. Cordoba, The Kakeya maximal function and the spherical
summation multipliers, Amer. J. Math. 99(1) (1977), 1–22.
[Fa] K. J. Falconer, “The geometry of fractal sets”, Cambridge
Tracts in Mathematics 85, Cambridge University Press, Cam-
bridge, 1986.
[LT] I.  Laba and T. Tao, An x-ray transform estimate in Rn, Rev.
Mat. Iberoamericana 17(2) (2001), 375–407. DOI: 10.4171/RMI/298.
[Ma] P. Mattila, “Geometry of sets and measures in Euclidean
spaces”, Fractals and rectifiability, Cambridge Studies in Advanced
Mathematics 44, Cambridge University Press, Cambridge, 1995.
Department of Mathematics and Statistics
University of Helsinki
FI-00014 Helsinki
Finland
E-mail address: tuomas.orponen@helsinki.fi
Primera versio´ rebuda el 16 de novembre de 2012,
darrera versio´ rebuda el 31 de gener de 2013.


Non-existence of multi-line Besicovitch sets

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