On restricted families of projections in R^3
aa r X i v : . [ m a t h . C A ] J a n ON RESTRICTED FAMILIES OF PROJECTIONS IN R KATRIN FÄSSLER AND TUOMAS ORPONENA
BSTRACT . We study projections onto non-degenerate one-dimensional fami-lies of lines and planes in R . Using the classical potential theoretic approachof R. Kaufman, one can show that the Hausdorff dimension of at most / -dimensional sets B ⊂ R is typically preserved under one-dimensional fami-lies of projections onto lines. We improve the result by an ε , proving that if dim H B = s > / , then the packing dimension of the projections is almost surelyat least σ ( s ) > / . For projections onto planes, we obtain a similar bound, withthe threshold / replaced by . In the special case of self-similar sets K ⊂ R without rotations, we obtain a full Marstrand type projection theorem for one-parameter families of projections onto lines. The dim H K ≤ case of the resultfollows from recent work of M. Hochman, but the dim H K > part is new: withthis assumption, we prove that the projections have positive length almost surely. C ONTENTS
1. Introduction 21.1. Dimension estimates for general sets 41.2. A Marstrand type theorem for self-similar sets 52. Acknowledgements 63. General sets 63.1. Classical bounds 63.2. Beyond the classical bounds 93.3. Proof of Theorem 1.7: the details 104. Sets with additional structure 224.1. BLP sets 224.2. Self-similar sets 245. Further results 265.1. Product sets and projections onto lines on a cone 275.2. Another lower bound for general sets 28Appendix A. A discrete version of Frostman’s lemma 29Appendix B. Auxiliary results for curves 30References 33
Mathematics Subject Classification.
1. I
NTRODUCTION
The study of orthogonal projections has a long history in the field of geometricmeasure theory. The foundations were laid in the late thirties by A.S. Besicovitch,who established that the structural properties of sets of Hausdorff dimension onein R are reflected in the size of their orthogonal projections into lines through theorigin. For planar sets of arbitrary dimension, a major breakthrough appearedsome 15 years later, in the 1954 paper [9] by J.M. Marstrand. He demonstratedthat for sets B ⊂ R with dimension at most one, almost all projections have thesame dimension, while the assumption dim H B > guarantees that almost allprojections have positive length. In the present article, we aim for results of thisnature in R .In R d , d ≥ , there are at least ( d − natural generalisations of Marstrand’s the-orem. Namely, one may pick k ∈ { , , . . . , d − } and start asking questions aboutorthogonal projections onto k -dimensional subspaces. In this situation, it turnsout that the dimension of at most k -dimensional sets B ⊂ R d is preserved underalmost all projections, while dim H B > k suffices for positive k -dimensional mea-sure – but, again, only for almost all projections. The generalisation, publishedin 1975, is due to P. Mattila [10].The results are complete, save for the word "almost". A moment’s thoughtreveals that the word may not be entirely omitted, as it is easy to come up withexamples of sets B ⊂ R d (line segments, for instance) for which the dimensionof projections is less than dim H B for a few exceptional subspaces. This does notmean that results sharper than the ones by Marstrand and Mattila are not possible– and indeed they are: numerous such improvements have appeared since 1954.A particularly elegant one is due to R. Kaufman [7] from 1968: given < s < ,a family L of lines through the origin in R such that dim H { L ∩ S : L ∈ L} = s ,and a set B ⊂ R with dim H B < s , one may always find a line L ∈ L such thatthe projection of B into L has dimension dim H B . Moreover, the result is sharp inthe sense that it may fail if dim H B = s . This was shown by Kaufman and Mattila[8] in 1975: for any < s < , they managed to construct an s -dimensional set B ⊂ R , the dimension of the projections of which drops strictly below s for acertain s -dimensional family L = L B of lines through the origin.To sum up, in casual terms, the results mentioned so far, it has been known forquite some time that orthogonal projections preserve dimension almost surely,but the family of exceptional projections can be fairly large. A question that re-mains, to date, largely unanswered, can be phrased as follows: What is the struc-ture of exceptional sets (of subspaces)? For instance, the construction of Kaufmanand Mattila from 1975 essentially relies on the fact that both B and L B can be cho-sen freely, and so as to play well together: it is far from clear that a less carefullychosen L would consist entirely of exceptional lines for any s -dimensional set B .In R , the question is wide open, but in R some understanding is emerg-ing. For those interested in a philosophical reason for how R can possibly N RESTRICTED FAMILIES OF PROJECTIONS IN R be easier than R , it is that the full families of projections onto one- and two-dimensional subspaces in R are two-dimensional – they can both be naturallyparametrised by the unit sphere S – so one can ask non-trivial questions aboutone-dimensional subfamilies. In the plane, however, the full family of lines isonly one-dimensional, so the interesting questions necessarily concern subfam-ilies of fractional dimension, and, at the moment the research seems to be evendevoid of plausible conjectures.For the rest of the introduction – and indeed the paper – we will be con-cerned with projections in R onto one- and two-dimensional subspaces. Moreprecisely, we are interested in smooth one-dimensional subfamilies of the full(two-dimensional) families. For the moment, let us fix such a family L of one-dimensional subspaces (essentially the same considerations are relevant for fami-lies of two-dimensional subspaces, so we will not grant them a separate treatmentin this informal discussion). The smoothness is quantified here by parametrisingthe family L by a smooth path γ : U → S (here U ⊂ R is an open interval) so that L = ( ℓ θ ) θ ∈ U := { span( γ ( θ )) : θ ∈ U } . The union of the lines in L forms a surface S , and it turns out that the curvatureof S plays a crucial role in our investigation. The necessity of curvature for non-trivial results is easy to see: if S is completely flat, that is, contained in a singletwo-dimensional subspace V , then so are all the lines in L , and the projection ofthe one-dimensional set B = V ⊥ into each line in L is the singleton { } . Then L consists entirely of exceptional lines with respect to B , in a very strong sense.More generally, if S is contained in a countable union of two-dimensional sub-spaces, then L can be shown to be entirely exceptional for some one-dimensionalset B ⊂ R – again in the sense that the dimension of the projection of B into L iszero for every line L ∈ L .For one-dimensional families of planes, the "non-curved" situation is slightlymore subtle, and for low-dimensional families of k -dimensional subspaces in R d ,the subtlety increases still: nevertheless, the best possible projection results with-out curvature conditions are known for all pairs k < d , due to the work of M.Järvenpää, E. Järvenpää, T. Keleti, F. Ledrappier and M. Leikas, see [5] and [6].At any rate, if we are interested in progress towards a Marstrand type theoremfor L , or for one-dimensional families of two-dimensional subspaces, we need toassume some curvature. Stated in terms of the parametrising path γ , the condi-tion used in the present paper reads as follows: Definition 1.1 (Non-degenerate families) . Let U ⊂ R be an open interval, and let γ : U → S be a C -curve on the unit sphere in R satisfying the condition span( { γ ( θ ) , ˙ γ ( θ ) , ¨ γ ( θ ) } ) = R , θ ∈ U. (1.2)To each point γ ( θ ) , θ ∈ U , we assign the line ℓ θ = span( γ ( θ )) . Any family oflines ( ℓ θ ) θ ∈ U so obtained is called a non-degenerate family of lines . The orthogonal KATRIN FÄSSLER AND TUOMAS ORPONEN complements V θ := ℓ ⊥ θ form a one-dimensional family of planes. Any family ofplanes ( V θ ) θ ∈ U so obtained is called a non-degenerate family of planes .As we have indicated, we are interested in projections corresponding to non-degenerate families of lines and planes. For these, we use the following notation: Definition 1.3 (Projections ρ θ and π θ ) . If ( ℓ θ ) θ ∈ U is a non-degenerate family oflines associated with the curve γ : U → S , we write ρ θ : R → R for the orthogo-nal projection ρ θ ( x ) = γ ( θ ) · x. Thus, we interpret the projection onto the line ℓ θ spanned by γ ( θ ) as a subset of R .Given a non-degenerate family of planes ( V θ ) θ ∈ U , we denote by π θ : R → R theorthogonal projection onto the plane V θ , identified with R . Projection familiesof the form ( ρ θ ) θ ∈ U and ( π θ ) θ ∈ U will be referred to as non-degenerate families ofprojections . Remark . The terminology "non-degenerate families of projections" is also usedin the papers [5] and [6] mentioned above, where no curvature assumptions areimposed. So, our definition is more restrictive, despite the common name.1.1.
Dimension estimates for general sets.
Unless otherwise stated, ( ρ θ ) θ ∈ U and ( π θ ) θ ∈ U will always stand for non-degenerate families of projections onto linesand planes, respectively. When dim H B lies on certain intervals, dimension con-servation for non-degenerate families of projections can be proven directly usingthe classical ‘potential theoretic’ method pioneered by R. Kaufman in [7]. Thesebounds are the content of the following proposition. Proposition 1.5.
Let B ⊂ R be an analytic set. (a) If dim H B ≤ / , then dim H ρ θ ( B ) = dim H B almost surely. (b) If dim H B ≤ , then dim H π θ ( B ) = dim H B almost surely. Part (b) follows from [6, Proposition 3.2]. The proof of part (a) is standard:we include it mainly to identify the ‘enemy’ against which we have to combat inorder to obtain an improvement, but also because the proof contains certain sub-level estimates needed to prove Theorem 1.7. Before stating any further results,let us formulate a conjecture:
Conjecture 1.6.
In Proposition 1.5(a), the hypothesis dim H B ≤ / can be relaxed to dim H B ≤ . In part (b), the hypothesis dim H B ≤ can be relaxed to dim H B ≤ . We fall short of proving the conjecture in two ways: first, we are only able toobtain a non-trivial lower bound for the packing dimension dim p (see [12, §5.9]) ofthe projections, and, second, our bound is much weaker than the full dimensionconservation conjectured above (the first shortcoming has already been partiallyovercome in later work, see [13] and [16]). Our first main result is the following: Theorem 1.7.
Let B ⊂ R be an analytic set, and write s := dim H B . N RESTRICTED FAMILIES OF PROJECTIONS IN R (a) If s > / , there exists a number σ = σ ( s ) > / such that dim p ρ θ ( B ) ≥ σ almost surely. (b) If s > , there exists a number σ = σ ( s ) > such that dim p π θ ( B ) ≥ σ almost surely.Remark . The lower bounds for σ ( s ) and σ ( s ) given by the proof of Theorem1.7 are most likely not optimal and certainly not very informative. They are σ ( s ) ≥
12 + 12 · (2 s − s + 4 s − and σ ( s ) ≥ s − s − . For σ ( s ) we also have the easy bound σ ( s ) ≥ s/ , see Proposition 5.4. The twobounds are equal when s ≈ . .The proof of Theorem 1.7 involves analysing the (hypothetical) situation wherethe packing dimension of the projections of B drops in positively many directionsvery close to the ‘classical’ bounds given by Proposition 1.5. Building on thiscounter assumption, we extract a large subset of B with additional structure.This information is used to show that the projections of the subset must havefairly large dimension.1.2. A Marstrand type theorem for self-similar sets.
For self-similar sets in R without rotations, we are able to obtain some optimal results, including the partof Conjecture 1.6 concerning the projections ρ θ . The reason is that such sets K enjoy the following structural property. If π : R → V is the orthogonal projectiononto any plane V ⊂ R and ε > , there exists a compact subset ˜ K ⊂ K with dim H ˜ K ≥ dim H π ( K ) − ε such that the restriction π | ˜ K is bi-Lipschitz. Our secondmain result is the following: Theorem 1.9.
Let K ⊂ R be a self-similar set without rotations. (a) If ≤ dim H K ≤ , then dim H ρ θ ( K ) = dim H K almost surely. (b) If dim H K > , then ρ θ ( K ) has positive length almost surely. The phrase ’self-similar set without rotations’ means that the generating simil-itudes of K have the form ψ ( x ) = rx + w for some r ∈ (0 , and w ∈ R . The caseof self-similar sets with only ’rational’ rotations easily reduces to the one with norotations, but we are not able to prove Theorem 1.9 for arbitrary self-similar setsin R . Part (a) of Theorem 1.9 follows from recent work of M. Hochman [3], butthe methods of proof are very different. Part (b) is new. In addition to the struc-tural property of self-similar sets mentioned above, the proof of part (b) relieson an application of Theorem 1.7(b). Unfortunately, the structural property cancompletely fail for general sets B ⊂ R , see Remark 4.10, so Theorem 1.9 does notseem to admit further generalisation with our techniques.Specialising Theorem 1.9 to the family of lines foliating the surface of a verticalcone in R , one immediately obtains the following corollary: KATRIN FÄSSLER AND TUOMAS ORPONEN
Corollary 1.10.
Let K ⊂ R be an equicontractive self-similar set with dim H K > / .Then (cos θ ) · K + (sin θ ) · K + K has positive length for almost all θ ∈ [0 , . A self-similar set is called equicontractive , if all the generating similitudes haveequal contraction ratios. The proof can be found in Section 5. The corollary isclose akin to Theorem 1.1(b), in Y. Peres and B. Solomyak’s paper [18] with thechoice C λ = (cos λ ) · K + (sin λ ) · K , in the notation of [18]. The self-similar sets C λ treated in in [18] are not as specific in form as the ones in Corollary 1.10. On theother hand, the proof in [18] is based on the concept of transversality and, hence,requires that the sets C λ satisfy the strong separation condition for all λ ∈ (0 , .This is generally not the case with the sets C λ = (cos λ ) · K + (sin λ ) · K above.1.2.1. Notation.
Throughout the paper we will write a . b , if a ≤ Cb for someconstant C ≥ . The two-sided inequality a . b . a , meaning a ≤ C b ≤ C a ,is abbreviated to a ∼ b . Should we wish to emphasise that the implicit constantsdepend on a parameter p , we will write a . p b and a ∼ p b . The closed ball in R d with centre x and radius r > will be denoted by B ( x, r ) .2. A CKNOWLEDGEMENTS
We are grateful to Spyridon Dendrinos, Esa and Maarit Järvenpää, Pertti Mat-tila and David Preiss for stimulating discussions and encouragement. We alsowish to thank Esa and Maarit Järvenpää for their hospitality during our visit atthe University of Oulu. Finally, we are indebted to an anonymous referee formany comments, which certainly increased the accessibility of the paper.3. G
ENERAL SETS
In this section, we prove Proposition 1.5 and Theorem 1.7. It suffices to proveall ‘almost sure’ statements for any fixed compact subinterval I of the parameterset U . For the rest of the paper, we assume that this interval is I = [0 , .3.1. Classical bounds.
As we mentioned in the introduction, the Hausdorff di-mension of an analytic set B ⊂ R d with dim H B ≤ k is preserved under almostevery projection onto an k -dimensional subspace in R n . This was proved by J. M.Marstrand [9] in 1954 for d = 2 and k = 1 and by P. Mattila for general k < d in1975. In 1968, R. Kaufman [7] found a "potential theoretic" proof for Marstrand’sresult, using integral averages over energies of projected measures. It is a natu-ral point of departure for our studies to see if Kaufman’s method could be usedto prove dimension conservation for non-degenerate families of projections ontolines and planes in R . For appropriate ranges of dim H B – namely when dim H B is small enough – the answer is positive. This is the content of Proposition 1.5. N RESTRICTED FAMILIES OF PROJECTIONS IN R Proof of Proposition 1.5.
We discuss first part (a) of the proposition, which con-cerns projections onto lines. Since orthogonal projections are Lipschitz contin-uous, the upper bound dim H ρ θ ( B ) ≤ dim H B holds for every parameter θ . Toestablish the almost sure lower bound dim H ρ θ ( B ) ≥ dim H B , we use the stan-dard potential-theoretic method. Let t < dim H B ≤ / and find a positive andfinite Borel regular measure µ which is supported on B and whose t -energy isfinite, I t ( µ ) := Z B Z B | x − y | − t dµ ( x ) dµ ( y ) < ∞ . Such a measure exists by Frostman’s lemma for analytic sets, see [1]. For each θ ∈ [0 , , the push-forward measure µ θ = ρ θ♯ µ defined by µ θ ( E ) := µ ( ρ − θ ( E )) isa measure supported on ρ θ ( B ) . Our goal is to prove that R I t ( µ θ ) dθ < ∞ , whichimplies I t ( µ θ ) < ∞ and thus dim H ρ θ ( B ) ≥ t for almost every θ ∈ [0 , . UsingFubini’s theorem, we find Z I t ( µ θ ) dθ = Z Z ρ θ ( B ) Z ρ θ ( B ) | u − v | − t dµ θ ( u ) dµ θ ( v ) dθ = Z B Z B (cid:18)Z | ρ θ ( x − y ) | − t dθ (cid:19) dµ ( x ) dµ ( y ) . Z B Z B | x − y | − t dµ ( x ) dµ ( y ) = I t ( µ ) . The inequality follows from the next lemma combined with the linearity of ρ θ . Lemma 3.1.
Given t < / , the estimate Z | ρ θ ( x ) | − t dθ . t holds for all x ∈ S .Proof of Lemma 3.1. We consider the function
Π : [0 , × S → R , Π( θ, x ) := ρ θ ( x ) = x · γ ( θ ) . To prove the lemma, we fix x ∈ S and study the behaviour of θ Π( θ, x ) . Wenote that ∂ θ Π( θ, x ) = x · ˙ γ ( θ ) and ∂ θ Π( θ, x ) = x · ¨ γ ( θ ) . If Π( θ, x ) = ∂ θ Π( θ, x ) = 0 for some x ∈ S and θ ∈ [0 , , we infer that x isorthogonal to both γ ( θ ) and ˙ γ ( θ ) . If this happens, the second derivative ∂ θ Π( θ, x ) cannot vanish, because then x would be orthogonal to ¨ γ ( θ ) as well, and this isruled out by the non-degeneracy condition (1.2). We have now shown that Π( θ, x ) = ∂ θ Π( θ, x ) = 0 ⇒ ∂ θ Π( θ, x ) = 0 for ( θ, x ) ∈ [0 , × S . A compactness argument then yields a constant c > suchthat max {| Π( θ, x ) | , | ∂ θ Π( θ, x ) | , | ∂ θ Π( θ, x ) |} ≥ c, ( θ, x ) ∈ [0 , × S . (3.2) KATRIN FÄSSLER AND TUOMAS ORPONEN
Since Z | Π( θ, x ) | − t dθ = Z ∞ L ( { θ ∈ [0 ,
1] : | Π( θ, x ) | ≤ r − t } ) dr, we will need a uniform estimate for the L measures of the sub-level sets of θ Π( θ, x ) | . Such an estimate is provided for instance by [2, Lemma 3.3]: for every k ∈ N , there is a constant C k < ∞ so that for every interval I ⊂ R , every f ∈ C k ( I ) and every λ > , L ( { θ ∈ I : | f ( θ ) | ≤ λ } ) ≤ C k (cid:18) λ inf θ ∈ I | ∂ kθ f ( θ ) | (cid:19) k . (3.3)The next lemma, proved in Appendix B, shows that the mapping θ Π( θ, x ) canonly have finitely many zeros on [0 , . Lemma 3.4.
Let γ : [0 , → S be a parameterised curve satisfying the condition (1.2) .Then there exists ε > such that for all x ∈ S the function θ → ρ θ ( x ) vanishes in atmost two points in every interval of length ε . Recall that in each of the finitely many points θ ∈ [0 , , where Π( θ , x ) = 0 ,either ∂ θ Π( θ , x ) = 0 or ∂ θ Π( θ , x ) = 0 . Now, the uniform continuity of Π andits partial derivatives guarantee that there exists an open ball U ( x ) centred at x with the following property: [0 , can be covered by a finite number of intervals I , . . . , I n ( x ) , for each of which there are numbers k i ∈ { , , } and c i > with inf θ ∈ I i | ∂ k i θ Π( y, θ ) | ≥ c i for all y ∈ U ( x ) . (3.5)Set c := min { c , . . . , c n ( x ) } . The sub-level set estimate (3.3) applied to this situa-tion yields L ( { θ ∈ I i : | Π( θ, y ) | ≤ λ } ) ≤ ( if k i = 0 , λ ≤ c C k i (cid:16) λc i (cid:17) ki if k i ∈ { , } for all i ∈ { , . . . , n ( x ) } and y ∈ U ( x ) . Hence, there exists a finite constant c ( x ) > such that L ( { θ ∈ [0 ,
1] : | Π( θ, y ) | ≤ λ } ) ≤ c ( x ) λ , for all y ∈ U ( x ) and < λ ≤ . The sphere S can be covered by finitely many balls of the form U ( x ) , so L ( { θ ∈ [0 ,
1] : | Π( θ, x ) | ≤ λ } ) . λ , for all x ∈ S and < λ ≤ . (3.6)Finally, we obtain Z | Π( θ, x ) | − t dθ = Z ∞ L ( { θ ∈ [0 ,
1] : | Π( θ, x ) | ≤ r − t } ) dr . Z ∞ r − t dr, where the right hand side is finite by the assumption t < / . (cid:3) N RESTRICTED FAMILIES OF PROJECTIONS IN R It remains to prove part (b) of the proposition. The result follows directly from[6, Proposition 3.2] or [5, Theorem 3.2], since the family ( π θ ) θ ∈ U is ‘full’ or ‘non-degenerate’ in the senses of [6] and [5]. We will need the sub-level estimate (3.9)later, so we choose to include the proof. As in (a), we are reduced to proving theuniform bound Z | π θ ( x ) | − t dθ = Z ∞ L ( { θ ∈ [0 ,
1] : | π θ ( x ) | ≤ r − t } ) dr . t , x ∈ S , (3.7)valid for < t < . To see this, we note that the function F : [0 , × S → R , F ( θ, x ) := | π θ ( x ) | = d ( x, ℓ θ ) = 1 − ρ θ ( x ) can have at most second order zeros. Indeed, the formulae F ( θ, x ) = 1 − ( x · γ ( θ )) , ∂ θ F ( θ, x ) = − x · γ ( θ ))( x · ˙ γ ( θ )) ,∂ θ F ( θ, x ) = − x · ˙ γ ( θ )) − x · γ ( θ ))( x · ¨ γ ( θ )) (3.8)reveal that if F ( θ, x ) = 0 for some θ ∈ [0 , and x ∈ S , then x is parallel to γ ( θ ) .This implies that x · ˙ γ ( θ ) = 0 and x · ¨ γ ( θ ) = 0 since γ · γ = 1 ⇒ γ · ˙ γ = 0 ⇒ γ · ¨ γ = 0 . This means that ∂ θ F ( θ, x ) = 0 , by (3.8). Now, tracing the proof of Lemma 3.4, weconclude that the number of zeros of the function θ F ( θ, x ) is finite for x ∈ S .Since | F ( θ, x ) | = | π θ ( x ) | , the sub-level estimate (3.3) and the compactness of S yield L ( { θ ∈ [0 ,
1] : | π θ ( x ) | ≤ λ } ) . λ, x ∈ S . (3.9)This proves (3.7) for < t < . (cid:3) Beyond the classical bounds.
It can be read from the proof of Proposition1.5 above, why the potential theoretic method does not directly extend beyondthe dimension ranges ≤ dim H B ≤ / (for lines) and ≤ dim H B ≤ (forplanes). If x, y ∈ R are points such that ( x − y ) ⊥ γ ( θ ) and ( x − y ) ⊥ ˙ γ ( θ ) forsome θ ∈ (0 , , then both the mapping θ ρ θ ( x − y ) and its first derivative havea zero at θ = θ . This means that Z dθ | ρ θ ( x − y ) | t = ∞ (3.10)for any t > / . Now, if the whole set B ⊂ R is contained on the line perpen-dicular to γ ( θ ) and ˙ γ ( θ ) , then all the differences x − y , x, y ∈ B enjoy the sameproperty. Thus, Z I t ( ρ θ♯ µ ) dθ = ∞ for any t > / and for any Borel measure µ supported on B .For the projections π θ , the situation is not so clear-cut. Again, the direct po-tential theoretic approach fails, because if x − y ∈ ℓ θ = V ⊥ θ for some θ ∈ (0 , ,then (3.10) holds for any t > , with ρ θ replaced by π θ . But, this time, we do not know if one can construct a set B ⊂ R with dim H B > such that most of thedifferences x − y , x, y ∈ B , lie on the lines ℓ θ , θ ∈ [0 , . Thus, for all we know, it isstill possible that estimates of the form Z I s ( π θ♯ µ ) dθ . I t ( µ ) < ∞ (3.11)hold for < s < t and for suitable chosen measures µ supported on B .3.2.1. Proof of Theorem 1.7: a sketch.
Being unable to verify an estimate of the form(3.11) – and knowing its impossibility for projections onto lines – our proof takesa different road. We will now give a heuristic outline of the argument used in theproof of Theorem 1.7(b), before working out the details (the proof of Theorem1.7(a) is similar but slightly more technical). We start with the counter assump-tion that the dimension of the projections π θ ( B ) drops very close to one in pos-itively many directions θ ∈ [0 , . Using this and the non-degeneracy condition,we find two short, disjoint, compact subintervals I, J ⊂ [0 , with the followingproperties:(i) The dimension of the projections π θ ( B ) is very close to one for ‘almost all’parameters θ ∈ I ∪ J .(ii) The surface C I := [ θ ∈ I ℓ θ is ‘directionally separated’ from the lines ℓ θ , θ ∈ J , in the sense that if x, y ∈ C I , then x − y forms a large angle with any such line ℓ θ .The next step is to project the set B onto the planes V θ , θ ∈ I . Because of (i), weknow that the projections π θ are, on average, far from bi-Lipschitz. This impliesthe existence of many differences near the lines V ⊥ θ = ℓ θ , θ ∈ I . Building onthis information, we find a large subset ˜ B ⊂ B lying entirely in a small neigh-bourhood of C I . The closer the dimension of the projections π θ ( B ) drops to onefor θ ∈ I , the larger we can choose ˜ B . Then, we recall (ii) and observe that thedifferences x − y with x, y ∈ ˜ B are directionally far from the lines ℓ θ , θ ∈ J (at least if | x − y | is large enough). This means, essentially, that the restrictions π θ | ˜ B : ˜ B → R , θ ∈ J , are bi-Lipschitz and shows that the dimension of π θ ( B ) exceeds the dimension of ˜ B for θ ∈ J . If the dimension of ˜ B was taken closeenough to the dimension of B , we end up contradicting (i).3.3. Proof of Theorem 1.7: the details.
We will not discuss the proof of Theorem1.7(a) informally, since the general outline resembles so closely the one in theproof of Theorem 1.7(b). Our first aim is to reduce the proof of Theorem 1.7 toverifying a discrete statement, Theorem 3.18, which concerns sets and projectionsat a single scale δ > . To this end, we need some definitions. N RESTRICTED FAMILIES OF PROJECTIONS IN R Definition 3.12 ( ( δ, s ) -sets) . Let δ, s > , and let P ⊂ R be a finite δ -separatedset. We say that P is a ( δ, s ) -set, if it satisfies the estimate | P ∩ B ( x, r ) | . (cid:16) rδ (cid:17) s , x ∈ R , r ≥ δ. Here | · | refers to cardinality, but, in the sequel, it will also be used to denotelength in R and area in R . This should cause no confusion, since for any set A only one of the possible meanings of | A | makes sense.In a way to be quantified in the next lemma, ( δ, s ) -sets are well-separated δ -netsinside sets with positive s -dimensional Hausdorff content (denoted by H s ∞ ). Thisprinciple – a discrete Frostman’s lemma – is most likely folklore, but we could notlocate a reference for exactly the formulation we need. So, we choose to includea proof in Appendix A. Lemma 3.13 (Frostman) . Let δ, s > , and let B ⊂ R be any set with H s ∞ ( B ) =: κ > . Then there exists a ( δ, s ) -set P ⊂ B with cardinality | P | & κ · δ − s . As we have seen, the potential theoretic method cannot be used to improveProposition 1.5, because the projections onto planes (resp. lines) may have first(resp. second) order zeros. Such zeros lie on certain ‘bad lines’, the unions ofwhich form ‘bad cones’ in R . Let us establish notation for these objects. Definition 3.14 (Cones spanned by curves on S ) . Let γ : [0 , → S be a curve.If I ⊂ [0 , is a compact subinterval, we write C I ( γ ) := [ θ ∈ I span( γ ( θ )) ⊂ R . Two special cases of this definition are of particular interest:
Definition 3.15 (Bad lines and bad cones for projection families) . Let γ : U → S be a non-degenerate curve as in Definition 1.1, and let η : U → S be the curve η ( θ ) := γ ( θ ) × ˙ γ ( θ ) | γ ( θ ) × ˙ γ ( θ ) | . (a) A bad line for the projection family ( ρ θ ) θ ∈ U is any line of the form b θ := span( η ( θ )) ⊂ R , θ ∈ U. Unions of bad lines form bad cones , as in the previous definition: if I ⊂ [0 , is a compact subinterval, we write C ρI := C I ( η ) . (b) For the projection family ( π θ ) θ ∈ U , the bad lines have the form ℓ θ = span( γ ( θ )) . We also define the bad cones C πI := C I ( γ ) , I ⊂ [0 , . The definitions of bad lines and cones for ( ρ θ ) θ ∈ U and ( π θ ) θ ∈ U are closely relatedwith the zeros of the projections. For instance, x ∈ b θ ⇐⇒ x ⊥ γ ( θ ) and x ⊥ ˙ γ ( θ ) , where the right hand side is just another way of saying that θ ρ θ ( x ) = γ ( θ ) · x and θ ∂ θ ρ θ ( x ) = ˙ γ ( θ ) · x vanish simultaneously at θ = θ . In particular, if x ∈ b θ , then Z θ + εθ − ε dθ | ρ θ ( x ) | t = ∞ for any ε > and t > / . For the projections π θ , the situation is even simpler:the mapping θ
7→ | π θ ( x ) | has a (first order) zero at θ = θ , if and only if x ∈ ℓ θ .Now we can explain how the non-degeneracy hypothesis (1.2) is used in theproof of Theorem 1.7. It will ensure that if I, J ⊂ [0 , are appropriately chosenshort intervals, then the bad cones C ρI , C ρJ (in part (a)) or C πI , C πJ (in part (b)) ‘pointin essentially different directions’. This concept is captured by the next definition: Definition 3.16.
Let γ : [0 , → S be any curve, and let I, J ⊂ [0 , be disjointcompact subintervals. We write C I ( γ ) C J ( γ ) , if there is a constant c = c ( γ, I, J ) > with the following property. If x, y ∈ C I ( γ ) and ξ ∈ C J ( γ ) ∩ S = { γ ( θ ) : θ ∈ J } , then (cid:12)(cid:12)(cid:12)(cid:12) x − y | x − y | − ξ (cid:12)(cid:12)(cid:12)(cid:12) ≥ c. An equivalent way to state the condition is to say that there is a constant L = L ( γ, I, J ) < such that every orthogonal projection from C I ( γ ) to a line on C J ( γ ) is L -Lipschitz. The next lemma shows how to find intervals I, J ⊂ [0 , such that C I ( γ ) C J ( γ ) . Lemma 3.17.
Given a C curve γ : [0 , → S with nowhere vanishing tangent, supposethat θ , θ ∈ (0 , are such that γ ( θ ) / ∈ span( { γ ( θ ) , ˙ γ ( θ } ) . Then there exist ε , ε > such that C I C J with I = [ θ − ε , θ + ε ] , J = [ θ − ε , θ + ε ] , C I = C I ( γ ) and C J = C J ( γ ) . This result is rather intuitive; a rigorous proof is given in Appendix B. Now weare prepared to formulate a δ -discretised version of Theorem 1.7. Theorem 3.18.
Let s > , and let P ⊂ B (0 , be a ( δ, s ) -set with cardinality | P | ∼ δ − s .The following statements hold for δ > small enough. N RESTRICTED FAMILIES OF PROJECTIONS IN R (a) Suppose that
I, J ⊂ [0 , are intervals such that C ρI C ρJ . If s > / , there exist ε = ε ( s ) > and σ = σ ( s ) > / with the followingproperty. Suppose that E I ⊂ I and E J ⊂ J have lengths | E I | ≥ δ ε and | E J | ≥ δ ε . Then there exists a direction θ ∈ E I ∪ E J such that | ρ θ ( P ( δ )) | ≥ δ − σ . (b) Suppose that
I, J ⊂ [0 , are intervals such that C πI C πJ . If s > , there exist ε = ε ( s ) > and σ = σ ( s ) > with the followingproperty. Suppose that E I ⊂ I and E J ⊂ J have lengths | E I | ≥ δ ε and | E J | ≥ δ ε . Then there exists a direction θ ∈ E I ∪ E J such that | π θ ( P ( δ )) | ≥ δ − σ . Let us briefly explain how Theorem 1.7 follows from its δ -discretised variant.First, we note that in order to derive statements like Theorem 1.7 for the packingdimension of projections, it suffices to prove their analogues for the upper boxdimension dim B , defined by dim B R = lim sup δ → log N ( R, δ ) − log δ for bounded sets R ⊂ R d , where N ( R, δ ) is the least number of balls of radius δ required to cover R . This reduction is possible thanks to the following lemma. Lemma 3.19.
Let σ > , let µ be a Borel regular measure, and let B ⊂ R be a µ -measurable set such that µ ( B ) > , and |{ θ ∈ [0 ,
1] : dim p ρ θ ( B ) < σ }| > . Then there exists a compact set K ⊂ B with µ ( K ) > such that |{ θ ∈ [0 ,
1] : dim B ρ θ ( K ) < σ }| > . Proof.
The proof is the same as that of [15, Lemma 4.5], except for some obviouschanges in notation. (cid:3)
The statement also holds with the projections ρ θ replaced by π θ . Now, if Theo-rem 1.7 failed for dim p , there would exist an analytic set B ⊂ R with dim H B > s such that the projections of B have packing dimension less than σ ∈ { σ , σ } ina set of directions of positive measure. Then, we could find a Frostman measure µ inside B and apply Lemma 3.19 to B , µ and σ . The conclusion would be thatalso the dim B -variant of Theorem 1.7 has to fail in a set of directions of positivemeasure. Proof of Theorem 1.7.
We will now describe how to use Theorem 3.18(a) to provethe dim B -variant of Theorem 1.7(a). The deduction of Theorem 1.7(b) from The-orem 3.18(b) is analogous. To reach a contradiction, suppose that dim H B = s > / , but there is a positive length subset E ⊂ [0 , such that dim B ρ θ ( B ) < σ − c (3.20)for all θ ∈ E and some small constant c > . Here σ = σ ( s ) > / is the con-stant from Theorem 3.18. Fix a Lebesgue point θ ∈ E , and then choose anotherLebesgue point θ ∈ E such that η ( θ ) / ∈ span( { η ( θ ) , ˙ η ( θ ) } ) , (3.21)where η = γ × ˙ γ/ | γ × ˙ γ | . This is precisely where we need the non-degeneracyhypothesis. Lemma 3.22.
Let γ : U → S be a C curve satisfying the non-degeneracy condition (1.2) . Then the curve η : U → S , given by η := γ × ˙ γ | γ × ˙ γ | , fulfills the same condition, thatis, span { η ( θ ) , ˙ η ( θ ) , ¨ η ( θ ) } = R , (3.23) for every θ ∈ U . It follows from this lemma, which is proved in Appendix B, and from Lemma3.4 that for any given -plane W ⊂ R there are only finitely many choices of θ ∈ [0 , such that η ( θ ) ∈ W : indeed, if ¯ n is the normal vector of the plane W , Lemma 3.4 implies that the mapping θ η ( θ ) · ¯ n can only have a boundednumber of zeros θ ∈ [0 , . Now we may apply Lemma 3.17 to the path η : thus,we find disjoint compact intervals I ∋ θ and J ∋ θ with the property that C ρI C ρJ . This places us in a situation, where we can apply Theorem 3.18(a). Let ε > bethe number defined there, and let δ > be so small the lengths of E I := E ∩ I and E J := E ∩ J exceed δ ε . Then use Lemma 3.13 to find a ( δ, s ) -set P ⊂ B withcardinality | P | ∼ δ − s . From (3.20), we see that | ρ θ ( P ( δ )) | ≤ | ρ θ ( B ( δ )) | . δ − σ + c for δ > and θ ∈ E I ∪ E J . For δ > small enough, this is incompatible with theconclusion of Theorem 3.18(a). (cid:3) It remains to prove Theorem 3.18. The basic approach for both (a) and (b) isthe same, but (b) is slightly simpler from a technical point of view. This is whywe choose to give the proof of (b) first.
Proof of Theorem 3.18(b).
Recall that P ⊂ B (0 , is a ( δ, s ) -set of cardinality | P | ∼ δ − s . We make the counter assumption that | π θ ( P ( δ )) | < δ − σ (3.24) N RESTRICTED FAMILIES OF PROJECTIONS IN R for all θ ∈ E I ∪ E J . The constant σ ∈ (1 , s ) will be fixed as the proof progresses.In particular, (3.24) means that for θ ∈ E J , the projection π θ ( P ) can be covered by . δ − σ discs of radius δ > .For x, y ∈ R and θ ∈ E I ∪ E J , we define the relation x ∼ θ y as follows: x ∼ θ y ⇐⇒ x = y and | π θ ( x ) − π θ ( y ) | ≤ δ, We also write T I ( x, y ) := |{ θ ∈ E I : x ∼ θ y }| . Our first aim is to use (3.24) to find a lower bound for the quantity E := X x,y ∈ P T I ( x, y ) = Z E I |{ ( x, y ) ∈ P : x ∼ θ y }| dθ Fix θ ∈ E I and choose a minimal (in terms of cardinality) collection of disjointdiscs D , . . . , D M ( θ ) ⊂ R such that diam( D j ) = δ and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ∩ M ( θ ) [ j =1 π − θ ( D j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) & | P | ∼ δ − s . (3.25)Then (3.24) implies that M ( θ ) . δ − σ . Next, we discard all the discs D j such that | P ∩ π − θ ( D j ) | ≤ . This way only . δ − σ points are deleted from the left hand sideof (3.25), so the inequality remains valid for the remaining collection of discs, andfor small δ > . The point of the discarding process is simply to ensure that |{ ( x, y ) ∈ [ P ∩ π − θ ( D j )] : x ∼ θ y }| = | P ∩ π − θ ( D j ) | −| P ∩ π − θ ( D j ) | & | P ∩ π − θ ( D j ) | for the remaining discs D j . This in mind, we estimate E from below: E & Z E I X remaining D j | P ∩ π − θ ( D j ) | dθ ≥ Z E I M ( θ ) X remaining D j | P ∩ π − θ ( D j ) | dθ & | E I | · δ σ · | P | & δ σ + ε − s . (3.26)Our second aim is to show that (3.26) gives some structural information about P ,if ε and σ are small. For x ∈ P we define a ‘neighbourhood’ N ( x ) of x by N ( x ) := P ∩ ( x + C πI (2 δ )) . Recall that C πI was defined as the union of the lines ℓ θ , θ ∈ I , perpendicular to theplanes V θ . The reason for defining N ( x ) as we do is the following: if y ∈ P \ N ( x ) ,then y − x / ∈ C πI (2 δ ) , so that the difference y − x stays at distance > δ from any of the orthogonal complements of the planes V θ . In particular, it is not possible that x ∼ θ y for any parameter θ ∈ I , which implies that E = X x ∈ P X y ∈ N ( x ) T I ( x, y ) . (3.27)To connect the sizes of the neighbourhoods N ( x ) with (3.26), we need a universalestimate for T I ( x, y ) : Lemma 3.28.
Let x, y ∈ R be δ -separated points. Then T I ( x, y ) . δ | x − y | . Proof.
Apply the sub-level estimate (3.9) with λ = δ . (cid:3) Now we are equipped to search for a large set N ( x ) ⊂ P . Suppose that | N ( x ) | ≤ δ − s + ε for every x ∈ P , where ε = s ( s − s − . Write A j ( x ) := { y ∈ R : 2 j ≤ | y − x | ≤ j +1 } . Recalling (3.27) and using theinequality min { a, b } ≤ a − s b s , a, b ≥ , we estimate as follows: E = X x ∈ P X δ ≤ j ≤ X y ∈ A j ( x ) ∩ N ( x ) T I ( x, y ) . δ X x ∈ P X δ ≤ j ≤ − j min {| N ( x ) | , | P ∩ B ( x, j +1 ) |} . δ X x ∈ P X δ ≤ j ≤ − j min (cid:26) δ − s + ε , (cid:18) j δ (cid:19) s (cid:27) ≤ X x ∈ P X δ ≤ j ≤ δ ( − s + ε )(1 − s ) ∼ δ − s + ε (1 − s )+1 · log (cid:18) δ (cid:19) . So, assuming that | N ( x ) | ≤ δ − s + ε , we can combine the bound above with (3.26)to conclude that δ σ + ε − s . δ − s + ε (1 − s )+1 · log (cid:18) δ (cid:19) . Since the implicit constants are independent of δ > , this shows that either(i) There exists a point x ∈ P with | N ( x ) | ≥ δ − s + ε , or(ii) σ + ε ≥ ε (1 − s ) .The proof of Theorem 3.18(b) nears its end. Our next lemma will show that theprojections π θ | N ( x ) , θ ∈ J , are essentially bi-Lipschitz, so the counter assumption N RESTRICTED FAMILIES OF PROJECTIONS IN R | π θ ( P ( δ )) | < δ − σ for θ ∈ J will force the inequality | N ( x ) | . δ − σ . In case (i)holds, this shows that σ ≥ s − ε = 1 + ( s − s − . If (i) fails, we conclude from (ii) that σ + ε ≥ ε (1 − s ) = 1 + ( s − s − . (3.29)Either way, Theorem 3.18(b) is true for any pair ( σ , ε ) satisfying (3.29).It remains to state and prove the bi-Lipschitz lemma. In order to make the samelemma useful in the proof of Theorem 3.18(a), we state a slightly more generalversion than we would need here. Lemma 3.30.
Assume that γ : [0 , → S is a curve, and C I C J for some intervals I, J ⊂ [0 , , where C I = C I ( γ ) and C J = C J ( γ ) . Let x ∈ R , τ > . Then, there existsa constant C ≥ , depending only on γ , I and J , such that whenever y, y ′ ∈ B (0 , satisfy y, y ′ ∈ x + C I ( δ τ ) and | y − y ′ | ≥ Cδ τ , then (cid:12)(cid:12)(cid:12)(cid:12) y − y ′ | y − y ′ | − ξ (cid:12)(cid:12)(cid:12)(cid:12) & γ,I,J , ξ ∈ C J ∩ S . (3.31) Proof.
Let c > be the constant from the definition of C I ( γ ) C J ( γ ) : thus, if u, v ∈ x + C I , then (cid:12)(cid:12)(cid:12)(cid:12) u − v | u − v | − ξ (cid:12)(cid:12)(cid:12)(cid:12) ≥ c for any vector ξ ∈ C J ∩ S . Suppose that y, y ′ ∈ B (0 , satisfy the hypotheses ofthe lemma, and find y , y ′ ∈ x + C I such that | y − y | ≤ δ τ and | y ′ − y ′ | ≤ δ τ . Notethat the points y and y ′ are at least Cδ τ / apart for C ≥ , and (cid:12)(cid:12)(cid:12)(cid:12) y − y ′ | y − y ′ | − ξ (cid:12)(cid:12)(cid:12)(cid:12) ≥ c − (cid:12)(cid:12)(cid:12)(cid:12) y − y ′ | y − y ′ | − y − y ′ | y − y ′ | (cid:12)(cid:12)(cid:12)(cid:12) , ξ ∈ C J ∩ S . (3.32)To estimate the negative term, consider the mapping b : R \ B (0 , Cδ τ / → S ,defined by b ( x ) = x/ | x | . Choosing C large enough, the mapping b can be made L -Lipschitz with L ≤ cδ − τ / , so (cid:12)(cid:12)(cid:12)(cid:12) y − y ′ | y − y ′ | − y − y ′ | y − y ′ | (cid:12)(cid:12)(cid:12)(cid:12) = | b ( y − y ′ ) − b ( y − y ′ ) |≤ cδ − τ | y − y | + | y ′ − y ′ | ) ≤ c . This and (3.32) give (3.31). (cid:3)
In the proof of Theorem 3.18(b), we apply the lemma with the non-parallelbad cones C πI and C πJ and with τ = 1 : let C & γ,I,J be the constant appearing in the statement of the lemma. If option (i) above is realised, we choose a Cδ -net ˜ P ⊂ N ( x ) . Then | ˜ P | & δ − s + ε , and the angle between any difference y − y ′ , y, y ′ ∈ ˜ P , and any line ℓ θ = V ⊥ θ , θ ∈ J , is bounded from below by a constant.This means that the restrictions π θ | ˜ P are bi-Lipschitz, so | π θ ( ˜ P ( δ )) | & δ − s + ε forany θ ∈ J . The proof of Theorem 3.18(b) is now completed in the manner wedescribed above. (cid:3) Next, we turn to the proof of Theorem 3.18(a). The structure will be familiar,but there are some additional steps to take.
Proof of Theorem 3.18(a).
All the way down to the lower energy estimate (3.26) theargument follows the proof of Theorem 1.7(b) with the obvious changes π θ ρ θ , ε ε , σ σ , and choosing the sets D , . . . , D M ( θ ) as δ -intervals in R rather than δ -discs in R .The analogue of (3.27) is E & δ σ + ε − s . (3.33)The first essential difference appears in the definition of the ‘neighbourhoods’ N ( x ) , x ∈ P . This time N ( x ) := P ∩ ( x + C ρI ( δ τ )) , where τ ∈ (0 , / is a parameter to be chosen soon. Recall that C ρI is the unionof the bad lines b θ , θ ∈ I , spanned by the vectors γ ( θ ) × ˙ γ ( θ ) . Contrary to whatwe did in part (b), if a point y ∈ P stays away from a neighbourhood N ( x ) ,we may not conclude that x θ y for all θ ∈ I . Instead, the event y / ∈ N ( x ) signifies that the mapping θ ρ θ ( x − y ) does not have a second order zeroon the interval I . Consequently, we have an improved estimate for T I ( x, y ) . An‘improved estimate’ means an improvement over the following universal bound,analogous to the one in Lemma 3.28: Lemma 3.34.
Let x, y ∈ R be δ -separated points. Then T I ( x, y ) . (cid:18) δ | x − y | (cid:19) / . Proof.
Apply the sub-level estimate (3.6) with λ = δ . (cid:3) Lemma 3.35.
Suppose that ≤ τ < , and x, y ∈ R satisfy y − x / ∈ C ρI ( δ τ ) . Then T I ( x, y ) . δ − τ . Proof.
The condition y − x / ∈ C ρI ( δ τ ) is another way of saying that d ( y − x, b θ ) ≥ δ τ for all bad lines b θ = span { γ ( θ ) × ˙ γ ( θ ) } ⊂ C ρI . Now, note that the distance of N RESTRICTED FAMILIES OF PROJECTIONS IN R a vector z from b θ equals the length of the projection ˜ π θ ( z ) of z onto the plane b ⊥ θ = span( { γ ( θ ) , ˙ γ ( θ ) } ) . Hence " (( x − y ) · γ ( θ )) + (cid:18) ( x − y ) · ˙ γ ( θ ) | ˙ γ ( θ ) | (cid:19) / = | ˜ π θ ( x − y ) | ≥ δ τ . Since | ˙ γ ( θ ) | is bounded from below on I , and τ < , we may infer that | ρ θ ( x − y ) | ≤ δ = ⇒ | ∂ θ ρ θ ( x − y ) | & δ τ . This implies that the set { θ ∈ I : | ρ θ ( x − y ) | < δ } consists of intervals I , . . . , I N around the zeros of θ ρ θ ( x − y ) on I , and possibly two intervals having acommon endpoint with I . We saw in Lemma 3.4 that the number of zeros of θ ρ θ ( x − y ) , x = y , on any compact subinterval of U is bounded by a constantindependent of x − y . So, in order to estimate the length of { θ ∈ I : | ρ θ ( x − y ) | < δ } – and the cardinality of T I ( x, y ) – it suffices to bound the lengths of the intervals I i . But the lower bound of the derivative ∂ θ ρ θ ( x − y ) readily shows that | I i | . δ − τ , which completes the proof of the lemma. (cid:3) Next, as in the proof of Theorem 3.18(b), we claim that the lower bound (3.33)forces a dichotomy: either ε and σ are large, or there exists a neighbourhood N ( x ) with cardinality | N ( x ) | ≥ δ − s + ε I . Here ε I = s (2 s − s + 4 s − − κ, where κ > is arbitrary (but so small that ε I > ). Let us first estimate E fromabove, assuming | N ( x ) | ≤ δ − s + ε I for every x ∈ P : E = X x ∈ P X y ∈ N ( x ) T I ( x, y ) + X x ∈ P X y ∈ P \ N ( x ) T I ( x, y ) =: S + S . The sum S is bounded using the universal bound in Lemma 3.34, combined withthe size estimate for | N ( x ) | : S . δ / X x ∈ P X δ ≤ j ≤ − j/ min {| N ( x ) | , | P ∩ B ( x, j +1 ) |} . δ / X x ∈ P X δ ≤ j ≤ − j/ min (cid:26) δ − s + ε I , (cid:18) j δ (cid:19) s (cid:27) ≤ X x ∈ P X δ ≤ j ≤ δ ( − s + ε I )(1 − s ) ∼ δ − s + ε I (1 − s )+1 / · log (cid:18) δ (cid:19) . To estimate S , we set τ = 1 / − ( ε I + ε J )(1 − s ) > , where ε J := 4 s s + 4 s − > , and apply Lemma 3.35 with this particular choice of τ : S ≤ X x ∈ P X y ∈ P \ N ( x ) δ − τ . δ − s +( ε I + ε J )(1 − s )+1 / . With our choices of parameters, we see that E = S + S . δ − s + ε I (1 − s )+1 / · log (cid:0) δ (cid:1) .Comparing this upper bound with (3.33), we conclude that one of the followingoptions must hold:(i) There exists x ∈ P with | N ( x ) | ≥ δ − s + ε I , or(ii) σ + ε ≥ / ε I (1 − s ) .Indeed, we obtained (ii) by assuming on the previous page that (i) fails for every x ∈ P . Now, (ii) would directly lead to a lower bound for σ (we will work outthe numbers soon), so (i) is the "hard" case. Thus, we momentarily assume that (i)holds and see where we end up. It is time to start using the information about thesize of the projections ρ θ ( P ( δ )) , θ ∈ J . Write ˜ P := N ( x ) , where | N ( x ) | ≥ δ − s + ε I .Since ˜ P ⊂ P , we know that | ρ θ ( ˜ P ( δ )) | ≤ δ − σ for θ ∈ E J . Consequently, if we set T J := |{ θ ∈ E J : x ∼ θ y }| and define E J := X x,y ∈ ˜ P T J ( x, y ) , the same argument that gave (3.33) yields the lower bound E J & δ σ + ε +2 ε I − s . (3.36)With this in mind, we set hunting for a large neighbourhood N J ( y ) := ˜ P ∩ ( y + C ρJ ( δ τ )) , y ∈ ˜ P .
The parameter τ > is the same as before. If all such neighbourhoods have size | N J ( y ) | ≤ δ − s + ε I + ε J , precisely the same argument as above shows that E J . δ − s +( ε I + ε J )(1 − s )+1 / · log (cid:18) δ (cid:19) . (3.37)Indeed, one only needs to observe that the bounds for T I ( x, y ) in Lemmas 3.34and 3.35 transfer without change to bounds for T J ( x, y ) . Comparing (3.36) and(3.37), we arrive at a familiar alternative:(i’) There exists y ∈ ˜ P with | N J ( y ) | ≥ δ − s + ε I + ε J , or(ii’) σ + ε ≥ / ε I + ε J )(1 − s ) − ε I . N RESTRICTED FAMILIES OF PROJECTIONS IN R Now the proof is nearly complete. The next step is to show that (i) and (i’) aremutually incompatible with our choices of ε I and ε J . Consequently, from ouralternatives, we will see that either (ii) holds, or (i) and (ii’) hold. Both optionswill lead to a lower bound for σ .To establish the incompatibility of (i) and (i’), we apply Lemma 3.30 with τ andthe non-parallel cones C ρI , C ρJ . Let C & γ,I,J be a constant, which appears in thelemma with these parameters. Assuming (i) and (i’), recalling the formulae for ε I and ε J , and using the fact that N J ( y ) is a ( δ, s ) -set, we have | N J ( y ) ∩ B ( y, Cδ τ ) | . C (cid:18) δ τ δ (cid:19) s = δ ( τ − s = δ − s/ − ( ε I + ε J )( s − / = δ κ ( s +1 / · δ − s + ε I + ε J ≤ δ κ ( s +1 / · | N J ( y ) | . This shows that no matter how large C is, for small enough δ > the set N J ( y ) cannot be contained in the ball B ( y, Cδ τ ) . So, if (i) and (i’) hold, and δ > is smallenough, we can find a point y ′ ∈ N J ( y ) ⊂ ˜ P ⊂ x + C ρI ( δ τ ) with | y − y ′ | > Cδ τ . (3.38)We infer from Lemma 3.30 that (cid:12)(cid:12)(cid:12)(cid:12) y − y ′ | y − y ′ | − ξ (cid:12)(cid:12)(cid:12)(cid:12) & γ,I,J , ξ ∈ C ρJ ∩ S . (3.39)On the other hand, we know that y ′ ∈ N J ( y ) ⊂ y + C ρJ ( δ τ ) , so there is a line b θ ⊂ C ρJ , θ ∈ J , such that d ( y − y ′ , b θ ) ≤ δ τ . If b θ is spanned by the unit vector ξ ∈ C ρJ ∩ S , one can combine (3.38) with elementary geometry to show that (cid:12)(cid:12)(cid:12)(cid:12) y − y ′ | y − y ′ | − ξ (cid:12)(cid:12)(cid:12)(cid:12) . C , as long as C ≤ δ − τ . This is incompatible with (3.39), if C is large enough (stilldepending only on γ , I and J ). We have established that (i) and (i’) cannot holdsimultaneously. Thus, if (i) holds, we may infer that also (ii’) holds, so σ + ε must satisfy the lower bound σ + ε ≥
12 + ( ε I + ε J ) (cid:18) − s (cid:19) − ε I = 12 + 12 · (2 s − s + 4 s − κ (cid:18) s (cid:19) But if (i) fails, we know that (ii) holds, and then σ + ε ≥
12 + ε I (cid:18) − s (cid:19) = 12 + 12 · (2 s − s + 4 s − − κ (cid:18) − s (cid:19) . Either way, making κ > small, we may choose ε ( s ) > and σ ( s ) > / as inTheorem 3.18(a). This completes the proof. (cid:3)
4. S
ETS WITH ADDITIONAL STRUCTURE
The dimension estimates obtained up to now for the projections of arbitrarysets B ⊂ R onto a non-degenerate family of lines in R are far from the optimalbound suggested by Conjecture 1.6. In this section, we restrict ourselves to aspecial class of sets B , for which we are able to prove stronger results – andindeed resolve part of Conjecture 1.6. The section has two parts. In the first one,we introduce BLP sets , a class of sets satisfying a strong structural hypothesis,and, in Theorem 4.7, we obtain sharp dimension estimates for such sets and non-degenerate families of projections onto lines. In the second part, we demonstratethat self-similar sets without rotations are BLP sets. Combined with Theorem 4.7,this fact yields a Marstrand type theorem for self similar sets: Theorem 1.9.4.1.
BLP sets.
We set off with two definitions.
Definition 4.1 (BLP sets) . A set B ⊂ R has the bi-Lipschitz property , BLP in short,if for any plane V ∈ G (3 , and ε > there exists a subset B V,ε ⊂ B such that • dim H B V,ε ≥ dim H π V ( B ) − ε , and • the restriction π V | B V,ε : B V,ε → V is bi-Lipschitz. Definition 4.2.
Let ℓ ∈ G (3 , . A set B ⊂ R stays non-tangentially off the line ℓ ( B ∠ ℓ for short) if there exists < α < such that X (0 , ℓ, α ) ∩ ( B − B ) = ∅ , where X ( y, ℓ, α ) := { x ∈ R : d ( x − y, ℓ ) < α | x − y |} is a cone with opening angle α around ℓ centered at y ∈ R .It will be useful to have various reformulations of this property at our dis-posal. We summarise them in the subsequent lemma, but omit the straightfor-ward proof. Lemma 4.3.
Let B ⊂ R and ℓ ∈ G (3 , . The following properties are equivalent: (1) B ∠ ℓ . (2) There exists < α < such that for all y ∈ B we have X ( y, ℓ, α ) ∩ B = ∅ . (3) The projection π V | B onto the plane V = ℓ ⊥ is bi-Lipschitz with the constant α from the definition of B ∠ ℓ . Let us now return to the projection family ( ρ θ ) θ ∈ U . The point of the definitionsabove is here: if B ⊂ R is a BLP set, θ ∈ U and ε > , one may find a subset B θ ,ε ⊂ B such that dim H B θ ,ε ≥ dim H π θ ( B ) − ε and B θ ,ε ∠ b θ , where b θ ∈ G (3 , is the ‘bad line’ spanned by the vector γ ( θ ) × ˙ γ ( θ ) and π θ is the projection onto b ⊥ θ . The next proposition explains why this is useful: N RESTRICTED FAMILIES OF PROJECTIONS IN R Proposition 4.4.
Let B ⊂ R be a set such that B ∠ b θ for some θ ∈ U . Then thereexists an open interval J ∋ θ such that the family ( ρ θ | B ) θ ∈ J is transversal in the senseof Peres and Schlag, see [17, Definition 2.7] .Remark . It would be unnecessarily cumbersome to recount here the full de-tails of Peres and Schlag’s framework. So, for the benefit of readers unfamil-iar with their definitions, we simply remark that Peres and Schlag’s paper dealswith generalised projections – parametrised families of continuous mappings froma compact space Ω to R k , satisfying certain properties. The essence of theseproperties is that they axiomatise those features of usual orthogonal projectionswhich are needed for the proofs of the classical dimension conservation resultsof Marstrand and Mattila. Consequently, an analogous dimension conservationtheorem holds for all families of generalised projections: in particular, the gen-eralised projections preserve, for almost all parameters, the dimension of any atmost k -dimensional Borel set in Ω . We wish to use this fact in the proof of The-orem 4.7 below, so, prior to the proof, we need to check that a certain family ofprojections, namely ( ρ θ | B ) θ ∈ J satisfies the axioms of the generalised projections.In this case, the task boils down to proving (4.6) below. Proof of Proposition 4.4.
Staying ‘non-tangentially off a line’ is an open property inthe following sense: if there exists < α < such that ( B − B ) ∩ X (0 , b θ , α ) = ∅ ,as we assume, then ( B − B ) ∩ X (0 , b θ , α/
2) = ∅ for θ in a small neighbourhood J ⊂ U of θ . Now, let θ ∈ J , and consider the projection π θ onto the plane V θ = b ⊥ θ .According to Lemma 4.3, the restriction π θ | B is bi-Lipschitz with constant α/ ,which means that " (( x − y ) · γ ( θ )) + (cid:18) ( x − y ) · ˙ γ ( θ ) | ˙ γ ( θ ) | (cid:19) / = | π θ ( x − y ) | ≥ α | x − y | for all x, y ∈ B . Taking J short enough, the quantity | ˙ γ ( θ ) | is bounded from belowby a constant c > for θ ∈ J . Thus, either (cid:12)(cid:12)(cid:12) ρ θ (cid:16) x − y | x − y | (cid:17)(cid:12)(cid:12)(cid:12) ≥ α or (cid:12)(cid:12)(cid:12) ∂ θ ρ θ (cid:16) x − y | x − y | (cid:17)(cid:12)(cid:12)(cid:12) ≥ cα . (4.6)for all x, y ∈ B , x = y . This means that J is an interval of transversality of order β = 0 for the projection family ( ρ θ | B ) θ ∈ J , in the sense [17, Definition 2.7]. (cid:3) Now we are prepared to prove the analogue of Theorem 1.9 for BLP sets.
Theorem 4.7.
Let B ⊂ R be a BLP set, and let ( ρ θ ) θ ∈ U be a non-degenerate family ofprojections in the sense of Definition 1.3. (a) If ≤ dim H B ≤ , then dim H ρ θ ( B ) = dim H B almost surely. (b) If dim H B > , and additionally dim p π V ( B ) = dim H π V ( B ) (4.8) for every plane V ∈ G (3 , , then ρ θ ( B ) has positive length almost surely. Proof.
We start with (a). According to Lemma 3.22, the family of lines ( b θ ) θ ∈ U is anon-degenerate one. Using Proposition 1.5(b), we see that dim H π θ ( B ) = dim H B (4.9)for almost every θ ∈ U , where π θ refers to the projection onto the plane V θ = b ⊥ θ .Let θ ∈ U be one of the parameters for which (4.9) holds, and fix ε > . Since B is a BLP set, we may choose a subset B θ ,ε ⊂ B such that dim H B θ ,ε ≥ dim H B − ε and B θ ,ε ∠ b θ . Then, we infer from Proposition 4.4 that there exists a smallinterval J ⊂ U containing θ such that the projections ( ρ θ ) θ ∈ J restricted to B θ ,ε are transversal. It follows from [17, Theorem 2.8] that dim H ρ θ ( B ) ≥ dim H ρ θ ( B θ ,ε ) = dim H B θ ,ε ≥ dim H B − ε for almost every θ ∈ J . Since (4.9) holds almost surely, we can run the sameargument for almost every θ ∈ U , proving that dim H ρ θ ( B ) ≥ dim H B − ε foralmost every θ ∈ U . Letting ε → concludes the proof of part (a).The proof of part (b) is similar, except that this time we resort to Theorem 1.7(b)instead of Proposition 1.5(b). Namely, if dim H B > , we infer from Theorem1.7(b) and the additional assumption (4.8) that dim H π θ ( B ) = dim p π θ ( B ) > for almost every θ ∈ U . Then, fixing almost any θ ∈ U and using the BLPproperty, we find a subset B θ ⊂ B such that dim H B θ > and B θ ∠ b θ . The restof the argument is the same a before, applying [17, Theorem 2.8] to the projections ρ θ , which are transversal restricted to the set B θ . (cid:3) Unfortunately, not all sets are BLP sets:
Remark . It is easy to construct a compact set K ⊂ R with dim H K = 1 suchthat dim H π V ( K ) = 0 (4.11)for a countable dense set of subspaces V ∈ G (3 , . Any such set K has the follow-ing property. Let V ∈ G (3 , , and let K be a subset of K such that the restriction π V | K is bi-Lipschitz. Then dim H K = 0 . Indeed, if π V | K is bi-Lipschitz, then π V | K is also bi-Lipschitz for all -planes V in a small G (3 , -neighbourhood of V . This means that dim H π V ( K ) ≥ dim H π V ( K ) = dim H K for all -planes V inan open subset of G (3 , , and now (4.11) forces dim H K = 0 .4.2. Self-similar sets.
In this section, we prove that self-similar sets without ro-tations in R satisfy the assumptions of Theorem 4.7. We start by setting some no-tation. Consider a collection { ψ , . . . , ψ q } of contracting similitudes ψ i : R → R .According to a result of Hutchinson [4] there exists a unique nonempty compactset K ⊂ R satisfying K = S qi =1 ψ i ( K ) . Such sets K are referred to as self-similarsets . If the generating similitudes of K have the form ψ i ( x ) = r i x + w i with < r i < and w i ∈ R , we call K a self-similar set without rotations . The factthat the mappings ψ i do not involve rotations will be used to guarantee that theprojection of K to an arbitrary plane is again self-similar. N RESTRICTED FAMILIES OF PROJECTIONS IN R Proposition 4.12.
Every self-similar set in R without rotations is a BLP set. Before presenting the proof, we recall some terminology from [14]. Rescalingthe translation vectors w i if necessary, we may assume that the similitudes ψ i , i ∈ { , . . . , q } , map the ball B (0 , ) into itself. Then, we set B = { B (0 , ) } andrefer to the recursively defined family B n := { ψ j ( B ) : B ∈ B n − , ≤ j ≤ q } as the collection of generation n balls of K associated with { ψ , . . . , ψ q } .The subset K V,ε to be constructed in the proof of Proposition 4.12 will be theattractor of a family of similitudes of the form { ψ B : B ∈ G} , where G is asuitably chosen collection of balls in S m ∈ N B m . Here, ψ B stands for a similitudeof the form ψ B = ψ i ◦ · · · ◦ ψ i n , mapping B (0 , ) to B = ψ i ◦ · · · ψ i n ( B (0 , )) .For a given B ∈ B n , the selection of ψ i , . . . , ψ i n may not be unique, but then anychoice is equally good for us. Observe that, for an arbitrary collection of balls G ⊆ S m ∈ N B m , the associated attractor is a subset of K . Also, since B was definedto consist of a single ball of diameter one, ψ B has contraction ratio diam( B ) .If { r , . . . , r q } are the contraction ratios of an IFS { ψ , . . . , ψ q } , then the similaritydimension of the associated attractor K is defined as the unique number s ≥ ,which solves the equation q X j =1 r sj = 1 . It is well known, see [4], that s = dim H K , provided that K exhibits a sufficientdegree of separation. One such condition is the very strong separation condition ,which, by definition, requires the generation balls of K to be disjoint. It is astronger requirement than the open set condition commonly used in literature, butwill be very convenient in the proof of Proposition 4.12. Proof of Proposition 4.12.
Let V ∈ G (3 , and ε > be arbitrary. The assumptionthat the similitudes ψ , . . . , ψ q generating the self-similar set K contain no rota-tions ensures that the set π V ( K ) is again self-similar. It is a subset of V , or, underthe customary identification, a subset of R , given by the IFS { ψ ,V , . . . , ψ q,V } with ψ j,V : R → R , ψ j,V ( x ) = r j x + π V ( w j ) . The corresponding collection of generation n balls will be denoted by B n,V . Ob-serve that the ball B (0 , ) in R is projected to the ball B (0 , ) in R , and hence B n,V comprises precisely the projections of the balls in B n .According to Lemma 3.4 in [14], we can for every ε > choose a self-similarset K V ⊂ π V ( K ) (depending on ε ) with dim H K V ≥ dim H π V ( K ) − ε satisfying thevery strong separation condition. In fact, the proof in [14] provides an IFS, whichgenerates the set K V and for which the generation balls are a subcollection B V of disjoint balls in B n,V , for some large n ∈ N . Moreover, we have X B ∈B V diam( B ) s = 1 (4.13)with s = dim H K V . Each ball B ∈ B V is the image of a ball in B n under theprojection π V . There might be several such balls in B n , but we just pick one ofthem. We denote by G the collection of balls in R obtained in this way. Sincethe balls in B V are disjoint, the balls in G are contained in disjoint well-separatedtubes perpendicular to V (and thus parallel to V ⊥ ). Also, (4.13) implies that X B ∈G diam( B ) s = 1 (4.14)with s = dim H K V . The set K V,ε ⊂ K , whose existence is claimed in the statementof the proposition, is obtained as the attractor of the IFS { ψ B : B ∈ G } . In otherwords, the balls in G form the generation balls of K V,ε . By (4.14) and the strongseparation condition, we have dim H K V,ε = dim H K V ≥ dim H π V ( K ) − ε. It remains to be established that the restriction of π V to K V,ε is bi-Lipschitz. Tothis end, we use the equivalent characterisation of this property in terms of conesas stated in Lemma 4.3. So far, we know that distinct balls in G are contained indisjoint closed tubes in direction V ⊥ . This allows us to find α > so that B ∩ X ( y, V ⊥ , α ) = ∅ for all y ∈ B ′ , (4.15)whenever B and B ′ are distinct balls in G . Then, it is a consequence of self-similarity that (4.15) holds with the same constant α for distinct generation n balls of K V,ε , for any n ∈ N . This is the content of the following lemma, a coun-terpart of which for sets in the plane is [14, Proposition 4.14]. Since the proof inhigher dimensions is completely analogous, we omit it here. Lemma 4.16.
Let n ∈ N be arbitrary and denote by G n the generation n balls of K V,ε .Then, whenever B and B ′ are distinct balls in G n , we have B ∩ X ( y, V ⊥ , α ) = ∅ for all y ∈ B ′ . Consequently, the restriction of π V to K V,ε is bi-Lipschitz with constant α , andthe proof of the proposition is complete. (cid:3) Finally, Theorem 1.9 follows by combining Theorem 4.7 with the BLP propertyof self-similar sets established in Proposition 4.12.5. F
URTHER RESULTS
This section contains further results concerning the projections onto a non-degenerate family of lines. It consists of two parts that, in specific situations,provide additional information to the dimension bounds obtained in Theorem1.7(a). In the first part, we consider a special non-degenerate family of lines,
N RESTRICTED FAMILIES OF PROJECTIONS IN R namely those foliating the surface of a cone, and the result obtained only appliesto sets of a certain product form. In the second part, we again return to arbitrarynon-degenerate families of lines and prove an explicit dimension bound for theassociated projections as stated in Remark 1.8.5.1. Product sets and projections onto lines on a cone.
Let K = K × K be aproduct set in R with K ⊂ R and K ⊂ R , and consider the curve γ : (0 , π ) → S (0 , √ given by γ ( θ ) = (cos( θ ) , sin( θ ) , . Then the lines ℓ θ := span( γ ( θ )) , θ ∈ (0 , π ) , foliate the surface of a vertical conein R , and the projections of K under ρ θ ( x ) := γ ( θ ) · x have a particularly simpleform: ρ θ ( K ) = ρ θ ( K × K ) = ℘ θ ( K ) + K , (5.1)where ℘ θ : R → R is the planar projection ℘ θ ( x, y ) = x cos θ + y sin θ . It is easyto verify that the curve γ (normalised by a constant) satisfies the non-degeneracycondition (1.2), so Theorem 1.9 holds for the projections ρ θ . Applying part (b)to the -fold product of an equicontractive self-similar set in R (which is a self-similar set in R ) and recalling (5.1) yields Corollary 1.10.As the first ‘further result’, we prove a variant of Theorem 1.7(a) for productsets K = K × K and the special family of projections ρ θ defined above. Proposition 5.2.
Let K = K × K ⊂ R , where K ⊂ R , K ⊂ R are analytic sets.Then dim H ρ θ ( K ) ≥ min { , dim H K } + dim H K for almost every θ ∈ (0 , π ) .Proof. Let µ and µ be positive Borel measures supported on K and K , re-spectively, such that I t ( µ ) < ∞ for some < t < min { dim H K , / } and I t ( µ ) < ∞ for some < t < dim H K . Then, with µ = µ × µ , we have Z π I t + t ( ρ θ♯ µ ) dθ = Z π (cid:18)Z | d ρ θ♯ µ ( r ) | | r | t + t − dr (cid:19) dθ ∼ Z π (cid:18)Z | ˆ µ ( rγ ( θ )) | | r | t + t − dr (cid:19) dθ = Z | ˆ µ ( r ) | (cid:18)Z π | ˆ µ ( r cos θ, r sin θ ) | dθ (cid:19) | r | t + t − dr. The inner integral is, by definition, the spherical average σ ( µ )( | r | ) of µ and anestimate of P. Mattila, see [11, Theorem 3.8], yields σ ( µ ) ( | r | ) . | r | − t I t ( µ ) . Here we needed the assumption t < / , which guarantees that t is within therange where the results from [11] apply. We may now conclude that Z π I t + t ( ρ θ♯ µ ) dθ . I t ( µ ) Z | ˆ µ ( r ) | | r | t − dr ∼ I t ( µ ) I t ( µ ) < ∞ , and thus I t + t ( ρ θ♯ µ ) < ∞ for almost every θ . This implies that dim H ρ θ ( K × K ) ≥ t + t for almost every θ ∈ (0 , π ) , and the proposition follows. (cid:3) Before moving on to other topics, we remark that, in light of (5.1), the followingconjecture is a weaker variant of Conjecture 1.6:
Conjecture 5.3.
Let K ⊂ R and K ⊂ R be analytic sets satisfying dim H K +dim H K ≤ . Then dim H ( ℘ θ ( K ) + K ) ≥ dim H K + dim H K for almost every θ ∈ (0 , π ) . Another lower bound for general sets.
In this section, we consider the gen-eral one-dimensional family of projections ( ρ θ ) θ ∈ U . Proposition 5.4. If K ⊂ R is an analytic set with ≤ dim H K ≤ , then dim p ρ θ ( K ) ≥ dim H K/ for almost every θ ∈ U . The proposition starts improving on the lower bound for σ ( s ) > / fromRemark 1.8 when dim H K = s ≈ . . Proof of Proposition 5.4.
Write dim H K =: s and assume < s ≤ . To reach acontradiction, suppose that there is a set E ⊂ U with positive length such that dim p ρ θ ( K ) < s/ for every θ ∈ E . Find two distinct Lebesgue points θ , θ ∈ E such that ( γ ( θ ) × ˙ γ ( θ )) · ˙ γ ( θ ) = 0 . Such points are given by the same argument as we used to obtain (3.21). Next,use continuity to find short open neighbourhoods
I, J ⊂ U of θ and θ such that | ( γ ( θ I ) × ˙ γ ( θ I )) · ˙ γ ( θ J ) | ≥ c > (5.5)for all ( θ I , θ J ) ∈ I × J ⊂ R . Then, consider the two-parameter family of projec-tions Π ( θ I ,θ J ) : R → R , ( θ I , θ J ) ∈ I × J , given by Π ( θ I ,θ J ) ( x ) := ( ρ θ I ( x ) , ρ θ J ( x )) = ( γ ( θ I ) · x, γ ( θ J ) · x ) . Using (5.5), one may check that this is a family of generalised projections satisfy-ing the framework of Peres and Schlag, see [17, Definitions 7.1 and 7.2]. Indeed,if x ∈ R is a unit vector such that, simultaneously, Π ( θ I ,θ J ) ( x ) = 0 and D Π ( θ I ,θ J ) ( x )( D Π ( θ I ,θ J ) ( x )) T ] = ( ˙ γ ( θ I ) · x ) + ( ˙ γ ( θ J ) · x ) , (5.6)then x is perpendicular to both planes span( { γ ( θ I ) , γ ( θ J ) } ) and span( { ˙ γ ( θ I ) , ˙ γ ( θ J ) } ) ,which implies that γ ( θ I ) · ( ˙ γ ( θ I ) × ˙ γ ( θ J )) = ( γ ( θ I ) × ˙ γ ( θ I )) · ˙ γ ( θ J ) , violating (5.5). Since we have not properly introduced the "generalised projec-tions" framework of Peres and Schlag (see Remark 4.5), we can only state thatchecking the requirements in [17, Definitions 7.1 and 7.2] amounts precisely to N RESTRICTED FAMILIES OF PROJECTIONS IN R verifying that Π ( θ I ,θ J ) ( x ) = 0 and the equation (5.6) cannot hold simultaneously –and this we have just done.Now [17, Theorem 7.3] implies that dim H Π ( θ I ,θ J ) ( K ) = s for almost every pair ( θ I , θ J ) ∈ I × J . On the other hand, Π ( θ I ,θ J ) ( K ) ⊂ ρ θ I ( K ) × ρ θ J ( K ) , so we obtain the estimate dim p ρ θ I ( K ) + dim p ρ θ J ( K ) ≥ dim H Π ( θ I ,θ J ) ( K ) = s for almost every pair ( θ I , θ J ) ∈ I × J . For such pairs ( θ I , θ J ) , we have either dim p ρ θ I ( K ) ≥ s/ or dim p ρ θ J ( K ) ≥ s/ , which means that | I || J | = | I × J | = |{ ( θ I , θ J ) : dim p ρ θ I ( K ) ≥ s or dim p ρ θ J ( K ) ≥ s }|≤ (cid:12)(cid:12)(cid:8) θ I ∈ I : dim p ρ θ I ( K ) ≥ s (cid:9)(cid:12)(cid:12) | J | + | I | (cid:12)(cid:12)(cid:8) θ J ∈ J : dim p ρ θ J ( K ) ≥ s (cid:9)(cid:12)(cid:12) . We may conclude that either (cid:12)(cid:12)(cid:8) θ I ∈ I : dim p ρ θ I ( K ) ≥ s (cid:9)(cid:12)(cid:12) ≥ | I | or (cid:12)(cid:12)(cid:8) θ J ∈ J : dim p ρ θ J ( K ) ≥ s (cid:9)(cid:12)(cid:12) ≥ | J | . However, since θ and θ were Lebesgue points of E , neither option is possible if I and J were chosen short enough to begin with. This contradiction completesthe proof. (cid:3) A PPENDIX
A. A
DISCRETE VERSION OF F ROSTMAN ’ S LEMMA
In this section, we prove Lemma 3.13. Let us recall the statement:
Proposition A.1.
Let δ > , and let B ⊂ R be a set with H s ∞ ( B ) =: κ > . Then,there exists a ( δ, s ) -set P ⊂ B with cardinality | P | & κ · δ − s .Proof. Without loss of generality, assume that δ = 2 − k for some k ∈ N and B ⊂ [0 , . Denote by D k the dyadic cubes in R of side-length − k . First, find all thedyadic cubes Q k ∈ D k which intersect B , and choose a single point x ∈ B ∩ Q k foreach Q k . The finite set so obtained is denoted by P . Next, modify P as follows.Consider the cubes in D k − . If one of these, say Q k − , satisfies | P ∩ Q k − | > (cid:18) d ( Q k − ) δ (cid:19) s , remove points from P ∩ Q k − , until the reduced set P ′ satisfies (cid:18) d ( Q k − ) δ (cid:19) s ≤ | P ′ ∩ Q k − | ≤ (cid:18) d ( Q k − ) δ (cid:19) s . Repeat this for all cubes Q k − ∈ D k − to obtain P . Then, repeat the procedure atall dyadic scales up from δ , one scale at a time: whenever P j has been defined,and there is a cube Q k − j − ∈ D k − j − such that | P j ∩ Q k − j − | > (cid:18) d ( Q k − j − ) δ (cid:19) s , remove points from P j ∩ Q k − j − , until the reduced set P ′ j satisfies (cid:18) d ( Q k − j − ) δ (cid:19) s ≤ | P ′ j ∩ Q k − j − | ≤ (cid:18) d ( Q k − j − ) δ (cid:19) s . (A.2)Stop the process when the remaining set of points, denoted by P , is entirely con-tained in some dyadic cube Q ⊂ [0 , . Now, we claim that for every point x ∈ P there exists a unique maximal dyadic cube Q x ⊂ Q such that ℓ ( Q x ) ≥ δ and | P ∩ Q x | ≥ (cid:18) d ( Q x ) δ (cid:19) s . (A.3)We only need to show that there exists at least one cube Q x ∋ x satisfying (A.3);the rest follows automatically from the dyadic structure. If x ∈ P , we have (A.3)for the dyadic cube Q x ∈ D k containing x . On the other hand, if x ∈ P \ P , thepoint x was deleted from P at some stage. Then, it makes sense to define Q x asthe dyadic cube containing x , where the ‘last deletion of points’ occurred. If thishappened while defining P j +1 , we have (A.2) with Q k − j − = Q x . But since thiswas the last cube containing x , where any deletion of points occurred, we see thatthat P ′ j ∩ Q x = P ∩ Q x . This gives (A.3).Now, observe that the cubes { Q x : x ∈ P } , • cover B , because they cover every cube in D k containing a point in P , andthese cubes cover B , • are disjoint, hence partition the set P .These facts and (A.3) yield the lower bound | P | = X | P ∩ Q x | & δ − s X d ( Q x ) s ≥ κ · δ − s . It remains to prove that P is a ( δ, s ) -set. For dyadic cubes Q ∈ D l with l ≤ k itfollows immediately from the construction of P , in particular the right hand sideof (A.2), that | P ∩ Q | ≤ (cid:18) d ( Q ) δ (cid:19) s . The statement for balls B ⊂ R with d ( B ) ≥ δ follows by observing that any suchball can be covered by ∼ dyadic cubes of diameter ∼ d ( B ) . (cid:3) A PPENDIX
B. A
UXILIARY RESULTS FOR CURVES
In this section, we prove Lemma 3.4, Lemma 3.17 and Lemma 3.22.
Proof of Lemma 3.4.
Consider the function
Π : [0 , × S → R , Π( θ, x ) := ρ θ ( x ) = γ ( θ ) · x, and let δ > be a constant such that max (cid:8) | Π( θ, x ) | , | ∂ θ Π( θ, x ) | , (cid:12)(cid:12) ∂ θ Π( θ, x ) (cid:12)(cid:12)(cid:9) ≥ δ, ( θ, x ) ∈ [0 , × S . (B.1) N RESTRICTED FAMILIES OF PROJECTIONS IN R Then, find ε > so that for all ( θ, x ) , ( θ ′ , x ) ∈ [0 , × S with | θ − θ ′ | < ε , we have max (cid:8) | Π( θ, x ) − Π( θ ′ , x ) | , | ∂ θ Π( θ, x ) − ∂ θ Π( θ ′ , x ) | , (cid:12)(cid:12) ∂ θ Π( θ, x ) − ∂ θ Π( θ ′ , x ) (cid:12)(cid:12)(cid:9) < δ. (B.2)We claim that the statement of the lemma holds for this choice of ε . Fix x ∈ S and let I ⊂ [0 , be an interval of length ε . To reach a contradiction, assume thatthere exist distinct points θ , θ , θ ∈ I such that Π( θ , x ) = Π( θ , x ) = Π( θ , x ) = 0 . Applying Rolle’s theorem to the function θ Π( θ, x ) , we conclude that thereare at least two points in I where also the derivative ∂ θ Π( · , x ) vanishes, and, byanother application of Rolle’s theorem, we find a point in I where also ∂ θ Π( · , x ) is zero. From (B.2) it follows that max (cid:8) | Π( θ, x ) | , | ∂ θ Π( θ, x ) | , (cid:12)(cid:12) ∂ θ Π( θ, x ) (cid:12)(cid:12)(cid:9) < δ for all θ ∈ I , which contradicts (B.1). (cid:3) Proof of Lemma 3.17.
Our goal is to find ε , ε > and L < such that | ρ θ ( x − y ) | ≤ L | x − y | for all x, y ∈ C I and θ ∈ C J , (B.3)where I = [ θ − ε , θ + ε ] and J = [ θ − ε , θ + ε ] . Elements in C I are of theform x = r x γ ( θ x ) with r x ∈ R and θ x ∈ I . As we will explain now, it is enoughto verify (B.3) for pairs x = r x γ ( θ x ) ∈ C I and y = r y γ ( θ y ) ∈ C I with r x , r y ≥ .Clearly, if (B.3) holds for all such pairs x, y then it also holds for pairs x, y with r x , r y ≤ . In case r x and r y have opposite signs, (B.3) will be valid with someconstants L ′ ∈ [ L, and ε ′ < ε . The precise condition on ε ′ > is that L < min θ x ,θ y ∈ I ′ γ ( θ x ) · γ ( θ y ) with I ′ := [ θ − ε ′ , θ + ε ′ ] . (B.4)This can be achieved by the continuity of γ , since γ ( θ ) · γ ( θ ) = 1 and L < .Now, fix x = r x γ ( θ x ) ∈ C I and y = r y γ ( θ y ) ∈ C I with r x r y ≤ . Then, assuming(B.3) for points on C I with the same sign, we have | ρ θ ( x − y ) | ≤ | ρ θ ( x ) | + | ρ θ ( y ) | ≤ L ( | x | + | y | ) ≤ dL | x − y | = L ′ | x − y | , where d = (cid:18) min θ x ,θ y ∈ I ′ γ ( θ x ) · γ ( θ y ) (cid:19) − / ≥ and L ′ = dL < by (B.4). The inequality | x | + | y | ≤ d | x − y | follows from ( | x | + | y | ) = r x + r y − r x r y ≤ d ( r x + r y − r x r y ( γ ( θ x ) · γ ( θ y )) = d | x − y | . It remains to prove (B.3) for points x = r x γ ( θ x ) and y = r y γ ( θ y ) with r x , r y ≥ and θ x , θ y ∈ I . Without loss of generality we assume that r x ≤ r y . The differen-tiability of γ at θ y yields | ρ θ ( x − y ) | = | [ r x γ ( θ x ) − r y γ ( θ y )] · γ ( θ ) |≤ | [ r x ˙ γ ( θ y )( θ x − θ y ) + ( r x − r y ) γ ( θ y )] · γ ( θ ) | + r x o ( | θ x − θ y | ) . Exploiting the assumption γ ( θ ) / ∈ span( { γ ( θ ) , ˙ γ ( θ ) } ) , we can find constants L < and ε , ε > such that | [ r x ˙ γ ( θ y )( θ x − θ y ) + ( r x − r y ) γ ( θ y )] · γ ( θ ) | ≤ L q r x | ˙ γ ( θ y ) | ( θ x − θ y ) + ( r x − r y ) for all θ x , θ y ∈ [ θ − ε , θ + ε ] =: I and θ ∈ [ θ − ε , θ + ε ] =: J . Given ε > , wecan make ε smaller still to ensure that the inequality | ˙ γ ( θ y ) || θ x − θ y | ≤ (1 + ε ) | γ ( θ x ) − γ ( θ y ) | holds whenever | θ x − θ y | ≤ ε . Choosing ε > small enough, inserting thisestimate to the upper bound for | ρ θ ( x − y ) | , and comparing the result with | x − y | = q r x + r y − r x r y γ ( θ x ) · γ ( θ y ) = q r x r y | γ ( θ x ) − γ ( θ y ) | + ( r x − r y ) , we see that | ρ θ ( x − y ) | ≤ L | x − y | for some L ∈ ( L , . We also need to know thatthe bounds implicit in o ( | θ x − θ y | ) can be chosen small in a manner dependingonly on ε , but this follows from the C regularity of γ . (cid:3) Proof of Lemma 3.22.
In order to establish (3.23) for all θ ∈ U , it is sufficient toshow ¨ η ( θ ) · ( η ( θ ) × ˙ η ( θ )) = 0 , for all θ ∈ U. (B.5)This condition means precisely that η ( θ ) , ˙ η ( θ ) and ¨ η ( θ ) are all of positive length, η ( θ ) and ˙ η ( θ ) are not parallel and hence span a plane, and this plane does notcontain ¨ η ( θ ) . In order to prove (B.5), we first evaluate η = γ × ˙ γ | γ × ˙ γ | = 1 | ˙ γ | γ × ˙ γ, ˙ η = (cid:18) | ˙ γ | (cid:19) ′ γ × ˙ γ + 1 | ˙ γ | γ × ¨ γ and ¨ η = (cid:18) | ˙ γ | (cid:19) ′′ γ × ˙ γ + 2 (cid:18) | ˙ γ | (cid:19) ′ γ × ¨ γ + 1 | ˙ γ | ˙ γ × ¨ γ + 1 | ˙ γ | γ × ... γ . Then, η × ˙ η = 1 | ˙ γ | ( γ × ˙ γ ) × ( γ × ¨ γ ) = 1 | ˙ γ | ( γ · ( ˙ γ × ¨ γ )) γ. Finally, ¨ η · ( η × ˙ η ) = 1 | ˙ γ | ( γ · ( ˙ γ × ¨ γ )) , which is non-vanishing, due to condition (1.2) for the curve γ . This concludes theproof of the lemma. (cid:3) N RESTRICTED FAMILIES OF PROJECTIONS IN R R EFERENCES [1] L. C
ARLESON : Selected Problems on Exceptional Sets , Van Nostrand, 1967[2] M. C
HRIST : Hilbert transforms along curves. I. Nilpotent groups , Ann. of Math. (2) , Issue3 (1985), pp. 575–596[3] M. H
OCHMAN : On self-similar sets with overlaps and inverse theorems for entropy ,arXiv:1212.1873[4] J. H
UTCHINSON : Fractals and self-similarity , Indiana Univ. Math. J. (1981), pp. 713–747[5] E. J ÄRVENPÄÄ , M. J
ÄRVENPÄÄ AND
T. K
ELETI : Hausdorff dimension and non-degenerate families of projections , appeared electronically in J. Geom. Anal. (2013),DOI:10.1007/s12220-013-9407-8[6] E. J
ÄRVENPÄÄ , M. J
ÄRVENPÄÄ , F. L
EDRAPPIER AND
M. L
EIKAS : One-dimensional fami-lies of projections , Nonlinearity (2008), pp. 453–463[7] R. K AUFMAN : On Hausdorff dimension of projections , Mathematika (1968), pp. 153–155[8] R. K AUFMAN AND
P. M
ATTILA : Hausdorff dimension and exceptional sea of linear transfor-mations , Ann. Adad. Sci. Fenn., Ser. A 1 Math. (1975), pp. 387–392[9] J.M. M ARSTRAND : Some fundamental geometrical properties of plane sets of fractional dimen-sions , Proc. London Math. Soc. (3) (1954), pp. 257-302[10] P. M ATTILA : Hausdorff dimension, orthogonal projections and intersections with planes , Ann.Acad. Sci. Fenn. Ser A I Math , Issue 2 (1975), pp. 227–244[11] P. M ATTILA : Spherical averages of Fourier transforms of measures with finite energy; dimensionof intersections and distance sets
Mathematika , Issue 2 (1987), pp. 207–228[12] P. M ATTILA : Geometry of sets and measures in Euclidean spaces , Cambridge UniversityPress, 1995[13] D. O
BERLIN AND
R. O
BERLIN : Application of a Fourier restriction theorem to certain familiesof projections in R , preprint (2013), arXiv:1307.5039[14] T. O RPONEN
On the distance sets of self-similar sets
Nonlinearity, , Issue 6 (2012), pp.1919–1929[15] T. O RPONEN : On the Packing Dimension and Category of Exceptional Sets of Orthogonal Pro-jections , preprint (2012), arXiv:1204.2121[16] T. O
RPONEN : Hausdorff dimension estimates for some restricted families of projections in R ,preprint (2013), arXiv:1304.4955[17] Y. P ERES AND
W. S
CHLAG : Smoothness of projections, Bernoulli convolutions, and the dimen-sion of exceptions , Duke Math. J. (2) (2000), pp. 193–251[18] Y. P
ERES AND
B. S
OLOMYAK : Self-similar measures and intersections of Cantor sets , Trans.Amer. Math. Soc. , No. 10 (1998), pp. 4065–4087D
EPARTMENT OF M ATHEMATICS AND S TATISTICS , U
NIVERSITY OF H ELSINKI , P.O.B. 68, FI-00014 H
ELSINKI , F
INLAND
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