aa r X i v : . [ m a t h . C A ] M a y ORTHOGONAL PROJECTIONS OF DISCRETIZED SETS
WEIKUN HE
Abstract.
We generalize Bourgain’s discretized projection theorem to higherrank situations. Like Bourgain’s theorem, our result yields an estimate for theHausdorff dimension of the exceptional sets in projection theorems formulatedin terms of Hausdorff dimensions. This estimate complements earlier resultsof Mattila and Falconer. Introduction
Fractal properties of orthogonal projections of subsets the Euclidean space havebeen intensively studied in fractal geometry (See the survey [7] for history and re-cent development). One of the fundamental problems asks for lower bounds on thesize of the projections of a given set to different directions. Since, in general, we donot expect the projection to be large in every direction, we ask more precisely tobound from above the size of the set of exceptional directions where an exceptionaldirection means a subspace onto which the projection is small. In this problem,the notion of size varies according to the context. For example, in a fractal geo-metric context, it is often the Lebesgue measure or the Hausdorff dimension. In adiscretized setting, we measure the size of a set by its covering number by δ -ballswhere δ > is the observing scale. In this setting, Bourgain established a dis-cretized projection theorem [2, Theorem 5] concerning rank one projections. Theprimary goal of the present paper is to generalize Bourgain’s result to higher rankprojections.1.1. Statement of the main result.
Let < m < n be positive integers. Let δ > . We endow R n with its usual Euclidean structure. For x ∈ R n , B ( x, δ ) standsfor the closed ball of radius δ and center x . Let A be a bounded subset of R n . Wewrite N δ ( A ) for the minimal number of balls of radius δ that is needed in order tocover A . This number represents the size of A at scale δ .We denote by Gr( R n , m ) the Grassmannian of m -dimensional subspaces in R n .For V ∈ Gr( R n , m ) , π V : R n → V stands for the orthogonal projection to V . If W ∈ Gr( R n , n − m ) , we define d ∡ ( V, W ) = | det( v , . . . , v m , w , . . . , w n − m ) | , where ( v , . . . , v m ) is an orthonormal basis of V and ( w , . . . , w n − m ) an orthonormalbasis of W and the determinant is with respect to any orthonormal basis of R n .For example d ∡ ( V, W ) = 0 if and only if V and W have nontrivial intersection. For ρ ≥ , we denote by V ∡ ( W, ρ ) the set of all V ∈ Gr( R n , m ) such that d ∡ ( V, W ) ≤ ρ .Recall that V ∡ ( W, is a submanifold of codimension in Gr( R n , m ) and belongsto the class of algebraic subvarieties known as Schubert cycles (see for example [10,Chapter 1, §5]).Our main result is the following. Theorem 1.
Let m < n be positive integers. Given < α < n and κ > , thereexists ǫ > such that the following holds for sufficiently small δ > . Let A be a Date : May 10, 2018. subset of R n contained in the unit ball B (0 , . Let µ be a probability measure on Gr( R n , m ) . Assume that (1) N δ ( A ) ≥ δ − α + ǫ ; (2) ∀ ρ ≥ δ, ∀ x ∈ R n , N δ ( A ∩ B ( x, ρ )) ≤ δ − ǫ ρ κ N δ ( A ); (3) ∀ ρ ≥ δ, ∀ W ∈ Gr( R n , n − m ) , µ ( V ∡ ( W, ρ )) ≤ δ − ǫ ρ κ . Then there is a set
D ⊂
Gr( R n , m ) such that µ ( D ) ≥ − δ ǫ and N δ ( π V ( A ′ )) ≥ δ − mn α − ǫ whenever V ∈ D and A ′ ⊂ A is a subset such that N δ ( A ′ ) ≥ δ ǫ N δ ( A ) . The m = 1 case is due to Bourgain [2]. For m ≥ , our result is new. Hypoth-esis (2) is a Frostmann type non-concentration condition on A . Without it we canhave example like A = B (0 , δ − αn ) , a ball of radius δ − αn , whose size is N δ ( A ) ≈ δ − α but whose projection to any V ∈ Gr( R n , m ) is of size N δ ( π V ( A )) ≈ δ − mn α . Hypothesis (3) is a non-concentration condition on the distribution of the subspace V . The set V ∡ ( W, ρ ) can be thought of as a ρ -neighborhood of the Schubert cycle V ∡ ( W, . For example if m = 1 , V lives in the projective space and (3) is asking µ to be not concentrated around any projective subspace. Note that the factor δ − ǫ in both (2) and (3) means the non-concentration property needs to be satisfied upto scale δ ǫ . So the parameter κ is about how good the assumptions are and ǫ isabout how much the assumptions can be relaxed and how good the conclusion is.1.2. Fractal geometric consequences.
Just like Bourgain’s discretized projec-tion theorem can be used to derive a projection theorem in terms of Hausdorffdimension [2, Theorem 4], Theorem 1 has the following consequence.
Theorem 2.
Let m < n be positive integers. Given < α < n and κ > , there is ǫ > such that the following is true. Let A ⊂ R n is an analytic set of dimension dim H ( A ) = α . Then the set of exceptional directions (cid:8) V ∈ Gr( R n , m ) | dim H ( π V ( A )) ≤ mn α + ǫ (cid:9) does not support any nonzero measure µ on Gr( R n , m ) with the following non-concentration property, ∀ ρ > , ∀ W ∈ Gr( R n , n − m ) , µ ( V ∡ ( W, ρ )) ≤ ρ κ . Endow the Grassmanian
Gr( R n , m ) with a rotation invariant Riemannian metricso that we can talk about Hausdorff dimension of subsets of Gr( R n , m ) . Theorem 2applied to a Frostman measure supported on the set of exceptional directions, weget Corollary 3.
Let m < n be positive integers. Given < α < n and κ > , thereis ǫ > such that the following holds. Let A ⊂ R n be an analytic set of dimension dim H ( A ) = α . Then dim H (cid:8) V ∈ Gr( R n , m ) | dim H ( π V ( A )) ≤ mn α + ǫ (cid:9) ≤ m ( n − m ) − κ. Note that m ( n − m ) is the dimension of Gr( R n , m ) . As κ → , we get(4) dim H (cid:8) V ∈ Gr( R n , m ) | dim H ( π V ( A )) ≤ mn dim H ( A ) (cid:9) ≤ m ( n − m ) − . This may be compared to estimates already known.
RTHOGONAL PROJECTIONS OF DISCRETIZED SETS 3
Theorem 4 (Mattila [17], Falconer [8], see also [19, §5.3]) . Let A ⊂ R n be ananalytic set of Hausdorff dimension dim H ( A ) = α . For any < s ≤ min { α, m } , dim H { V ∈ Gr( R n , m ) | dim H ( π V ( A )) < s } ≤ m ( n − m ) − (max { α, m } − s ); Compared to Theorem 4, the estimate (4) provides new information in the fol-lowing two situations:(i) (Projection to lines) m = 1 and dim H ( A ) ∈ ]0 , n − [ ,(ii) (Projection to hyperplanes) m = n − and dim H ( A ) ∈ ] n − − n − , n [ .For example, for n = 2 and m = 1 , the case treated by Bourgain [2], dim H (cid:8) θ ∈ Gr( R , | dim H ( π θ ( A )) ≤
12 dim H ( A ) (cid:9) = 0 , for all analytic sets A such that < dim H ( A ) < . This estimate is also obtainedby Oberlin [20] using different methods. Bourgain’s approach has the advantage ofgiving an estimate with the ǫ and κ terms, or in other words that for any c > , lim ǫ → sup A dim H (cid:8) θ ∈ Gr( R , | dim H ( π θ ( A )) ≤
12 dim H ( A ) + ǫ (cid:9) = 0 , where A ranges over all analytic sets with Hausdorff dimension between c and − c .Note also that Corollary 3 can be reformulated in a similar way.Theorem 2 can be combined with Remez-type inequalities to study restrictedfamily of projections. Instead of looking at projections to all subspaces, we re-strict our attention to a family of subspaces. The non-concentration property inTheorem 2 translates to a transversality condition on the family. In the followingcorollary, we assume the family to be analytic and not contained in any properSchubert cycle. Corollary 5.
Let m < n be positive integers. Given < α < n and κ > , there is ǫ > such that the following holds. Let p ≥ be an integer and Ω ⊂ R p a connectedopen subset. Let V : Ω → Gr( R n , m ) be a real analytic map. Let A ⊂ R n be ananalytic set of dimension dim H ( A ) = α . If for any W ∈ Gr( R n , n − m ) , there exists t ∈ Ω such that V ( t ) ⊕ W = R n , then for any relatively compact subset Ω ′ in Ω ,there exists a constant d = d ( V, Ω ′ ) > such that (5) dim H (cid:8) t ∈ Ω ′ | dim H ( π V ( t ) ( A )) ≤ mn α + ǫ (cid:9) ≤ p − dκ. If moreover V is polynomial then d is independent of Ω ′ and proportional to thedegree. The study of restricted family of projections started long ago and saw significantprogress recently. We refer the reader to, for example, [23, 13, 12, 9, 21, 6, 14, 22].The recent interest is focused on whether for almost all parameters t , the dimensionof the projection dim H ( π V ( t ) ( A )) is at least the minimum between dim H ( A ) , theoriginal dimension, and m , the dimension of the subspaces to which we project (seefor example [9, Conjecture 1.6]). Corollary 5 deals with a different but parallelquestion. Here we compare dim H ( π V ( t ) ( A )) to mn dim H ( A ) + ǫ . Understandably,the exceptional set is much smaller.1.3. Ergodic motivation.
In [3], Bourgain, Furman, Lindenstrauss and Mozesused Bourgain’s discretized projection theorem together with harmonic analysis toshow equidistributions of linear random walks on the torus. Our primary motiva-tion behind Theorem 1 resides also in this ergodic problem. In Bourgain-Furman-Lindenstrauss-Mozes theorem, there is technical assumption which is the proxi-mality. While a subgroup Γ ⊂ SL d ( Z ) acts on the torus, its transpose t Γ acts onFourier coefficients. Bourgain’s discretized projection theorem is used to study largeFourier coefficients under this action. By the theory of random matrix products, WEIKUN HE if Γ is proximal, then large random products in Γ behave like rank one projectionscomposed with rotations, if viewed at an appropriate scale. When Γ is not proxi-mal, they behave like rank p projections composed with rotations, where p ≥ isthe proximality dimension of the random walk. Thus, we hope Theorem 1 will beuseful for understanding the non-proximal situation.1.4. Strategy of the proof.
Now we describe an outline of the proof of Theorem 1.Fix integers < m < n and a real number < α < n . For ǫ > and boundedsubset A ⊂ R n we define the set of exceptional directions to be(6) E ( A, ǫ ) = { V ∈ Gr( R n , m ) | ∃ A ′ ⊂ A, N δ ( A ′ ) ≥ δ ǫ N δ ( A ) and N δ ( π V ( A ′ )) < δ − mn α − ǫ } . When there is no ambiguity, we omit the variable ǫ and write simply E ( A ) . Ourtask is to bound µ ( E ( A )) given the distribution µ of the subspaces. In order toprove Theorem 1 which says µ ( E ( A )) ≤ δ ǫ under the assumptions of the theorem,we prove instead that µ ( E ( A ′ )) ≤ δ ǫ for some subset A ′ of A . Theorem 6.
Let m < n be positive integers. Given < α < n and κ > , thereexists ǫ > such that the following holds for sufficiently small δ > . Let A be asubset of R n contained in the unit ball B (0 , . Let µ be a probability measure on Gr( R n , m ) . Assume (1) , (2) and (3) , then there exists A ′ ⊂ A such that µ ( E ( A ′ )) ≤ δ ǫ . This statement is seemingly weaker, but there is actually a rather formal ar-gument which allows to deduce Theorem 1 from Theorem 6. We will show thisimplication in Proposition 25.The proof of Theorem 6 starts with the special case where n = 2 m . Proposition 7.
Theorem 6 is true if n = 2 m . As in the m = 1 case in [2], this special case is proved using a sum-product theo-rem. For m > , we need the higher dimensional sum-product estimate establishedin [11] which we recall here. Below and throughout this paper, for subsets X, Y ofa linear space, we denote by X + Y their sumset : X + Y = { x + y | x ∈ X, y ∈ Y } . Theorem 8 ([11, Theorem 3]) . Let m be a positive integer. Given κ > and σ < m , there is ǫ > such that the following holds for δ > sufficiently small. Let A be a subset of the space of linear endomorphisms End( R m ) and X a subset of R m , assume that(i) A ⊂ B (0 , δ − ǫ ) ,(ii) ∀ ρ ≥ δ , N ρ ( A ) ≥ δ ǫ ρ − κ ,(iii) for any nonzero proper linear subspace W ⊂ R n , there is a ∈ A and w ∈ W ∩ B (0 , such that d ( aw, W ) ≥ δ ǫ .(iv) X ⊂ B (0 , δ − ǫ ) ,(v) ∀ ρ ≥ δ , N ρ ( X ) ≥ δ ǫ ρ − κ ,(vi) N δ ( X ) ≤ δ − σ − ǫ .Then, N δ ( X + X ) + max a ∈A N δ ( X + aX ) ≥ δ − ǫ N δ ( X ) . The proof of Proposition 7 follows closely that in [2]. The main idea is touse additive combinatorial tools such as the Balog-Szemerédi-Gowers theorem toreduce to the situation where A is a cartesian product X × X with X ⊂ R m . Thenprojections of X × X to subspaces of dimension m correspond exactly to the sum-product operations X + aX , a ∈ End( R m ) , in Theorem 8. Finally, Theorem 8shows that the projection gained a factor δ − ǫ in size compared to X which has half RTHOGONAL PROJECTIONS OF DISCRETIZED SETS 5 the dimension of A . A technical point appearing in this proof is that the set X ,which is roughly a projection of A , has to satisfy the non-concentration propertyrequire by Theorem 8. This is addressed in Lemma 27.Once we have Proposition 7 we would like to reduce other cases to it. First,using a simple induction, we show that Theorem 6 holds if m divides n . Proposition 9.
Let q ≥ be an integer. If Theorem 6 is true for n ′ = ( q − m and m then it is also true for n = qm and m . The proof of Proposition 9 goes roughly as follows. If Theorem 6 fails for n = qm and m with the set A . Then for a lot of V ∈ Gr( R n , m ) , the projection π V ( A ) issmall : N δ ( π V ( A )) ≤ δ − mn α − ǫ . This implies that the δ -neighborhood of a fiber of π V has a large intersection with A . This means that there is a n ′ -dimensional slice(of thickness δ ) of A which has a covering number ≥ δ − n ′ n α + ǫ . Now we can applyTheorem 6 with n ′ and m to this slice. The main technical issue appearing hereis to ensure that the slice has the correct non-concentration property and this isaddressed in Lemma 28.If m does not divide n and m < n , write n = qm + r with < r < m . We canreduce the ( n, m ) -case to the ( n, qm ) -case. Proposition 10.
Let < m < n be such that qm < n where q ≥ . If Theorem 6is true for n and m ′ = qm then it is also true for n and m . The idea is the following. Let V , . . . , V q be random m -planes distributed inde-pendently according to µ . Thanks to the non-concentration property of µ , the sum V = V + · · · + V q is a direct sum in well-spaced position with large probablity.Thus the size of the projection π V ( A ) is comparable to the product of the sizes of π V i ( A ) , i ∈ { , . . . , q } . Applying Theorem 6 with n and m ′ = qm to A and thedistribution of V , we conclude that with large probability, π V ( A ) has size largerthan δ − qmn α − qǫ and hence for some i , π V i ( A ) has size larger than δ − mn α − ǫ .If m does not divide n and m > n , write n = q ( n − m ) + r with < r ≤ n − m and we reduce to the ( n, r ) -case. Proposition 11.
Let < m < n be such that n = q ( n − m ) + r where q ≥ and < r ≤ n − m . If Theorem 6 is true for n and m ′ = r then it is also true for n and m . This last reduction is the trickiest one. We are in a dual situation to the previousone. Again let V , . . . , V q be random m -planes distributed independently accordingto µ . This time we consider the intersection instead of the sum of these subspaces.With large probability, the intersection V = V ∩ · · · ∩ V q has dimension r . Thus,we can apply Theorem 6 with n and m ′ = r to π V ( A ) . Then the main task is torelate the size of π V ( A ) to those of π V i ( A ) . We would like to say that π V ( A ) beinglarge implies one of the π V i ( A ) must be large as well. However, this is not true ingeneral. It becomes true only if we know that no fiber of π V has large intersectionwith A (larger than δ − n − rn α − ǫ ). This relation is proved in Proposition 34 using arefinement (Lemma 37) of a combinatorial projection theorem due to Bollobás andThomason [1]. It remains to treat the case where there is a fiber of π V having largeintersection with A or, in other words, the case where A has a ( n − r ) -dimensionalslice with covering number ≥ δ − n − rn α − ǫ . The idea is to apply a projection theoremto this slice. Since it has a very large size, we achieve this even without a non-concentration property (Proposition 29).Now let us see how to prove Theorem 6 by putting these propositions together. Proof of Theorem 6.
Propositions 7 and 9 imply the theorem for all pairs ( n, m ) such that m divides n . Consider the following order on pairs of positive integers of WEIKUN HE the form ( n, m ) , < m < n . We say ( n, m ) ≺ ( n ′ , m ′ ) if ( n, min( m, n − m ) , m ) issmaller than ( n ′ , min( m ′ , n ′ − m ′ ) , m ′ ) for the lexicographical order.If the theorem were false then let ( n, m ) be a ≺ -minimal pair for which thetheorem fails. We know that m does not divide n . If m < n then write n = qm + r with < r < m . We have ( n, qm ) ≺ ( n, m ) . Hence Proposition 10 contradictsthe minimality of ( n, m ) . Otherwise m > n , then write n = q ( n − m ) + r with < r ≤ n − m . We have ( n, r ) ≺ ( n, m ) and then Proposition 11 contradicts theminimality of ( n, m ) . (cid:3) Acknowledgements.
This work is part of my PhD thesis conducted under thesupervision of Emmanuel Breuillard and Péter Varjú. I am greatly indebted to myadvisors for their help. I am also grateful to Nicolas de Saxcé for stimulating con-versations and to Julien Barral, Yichao Huang and Elon Lindenstrauss for helpfulcomments. 2.
Preliminaries
In this section we introduce notation that will be used throughout the paper,then provide some elementary estimates about the Grassmannian and finally recallsome tools from additive combinatorics.2.1.
Notation and basic definitions.
Throughout this paper, m and n will bepositive integers that denote dimensions. For any finite set A , we denote by | A | itscardinality. We endow R n with its usual Euclidean structure. We denote by O( n ) the orthogonal group on R n , by λ the Lebesgue measure on R n and by Gr( R n , m ) the Grassmannian of m -dimensional subspaces of R n . For a linear subspace V ⊂ R n ,denote by π V the orthogonal projection onto V . Recall that there is a uniqueEuclidean structure on each of the exterior powers V m R n for which the standardbasis is a orthonormal basis.Let δ > be a real number that we will refer to as the scale. For a point x ∈ R n ,we write B ( x, δ ) or x ( δ ) to denote the closed ball of radius δ centered at x . Let A be a bounded subset of R n . We denote by A ( δ ) the closed δ -neighborhood of A .When we observe a set A at scale δ , there are several quantities describing thesize of A . They differ one from another at most by a constant factor dependingonly on n . The first one is the external covering number by δ -balls (also knownas the metric entropy), denoted by N δ ( A ) . It is defined as the minimal number ofpoints x , . . . , x N such that the balls x ( δ )1 , . . . , x ( δ ) N cover A . Let ˜ A be a maximal δ -separated subset of A . Its cardinality also reflects the size of A at scale δ . Wecan also consider the Lebesgue measure λ ( A ( δ ) ) of the δ -neighborhood of A . Hereis a relation between these quantities. Lemma 12.
Let δ > and let A be a bounded subset of R n . Let ˜ A be a maximal δ -separated subset of A . Then (7) N δ ( A ) ≤ | ˜ A | ≤ N δ ( A ) ≤ N ( B (0 , N δ ( A ) , and | ˜ A | ≤ λ ( A ( δ ) ) λ ( B (0 , δ )) ≤ n N δ ( A ) . As a consequence, N δ ( A ( δ ) ) ≪ n N δ ( A ) . It is sometimes useful to change scale. Clearly, N δ ( A ) is nonincreasing in δ .Conversely, for all δ ′ ≥ δ , we have(8) N δ ( A ) ≪ n (cid:16) δ ′ δ (cid:17) n N δ ′ ( A ) . RTHOGONAL PROJECTIONS OF DISCRETIZED SETS 7 If f : R m → R n is a linear map with k f k ≤ K where K ≥ , or more generallyif f : A → R n is K -Lipschitz, we have(9) N δ ( f A ) ≪ n K n N δ ( A ) . When we want intersect two discretized sets
A, B ⊂ R n , we shall take the δ -neighborhood of at least one of the sets before intersecting. Note that N δ ( A ( δ ) ∩ B ( δ ) ) can be large while at the same time A ∩ B is empty. The same goes with A (2 δ ) ∩ B (2 δ ) and A ( δ ) ∩ B ( δ ) . However, we have(10) N δ ( A (2 δ ) ∩ B ) ≪ n N δ ( A ( δ ) ∩ B ( δ ) ) ≪ n N δ ( A ∩ B (2 δ ) ) . Distance on the Grassmannian.
For linear subspaces
V, W of R n , we define d ∡ ( V, W ) = k v ∧ · · · ∧ v r ∧ w ∧ · · · ∧ w s k where ( v , . . . , v r ) is an orthonormal basis of V and ( w , . . . , w s ) an orthonormalbasis of W . It is a distance when restricted to the projective space Gr( R n , butonly in this case. For example, d ∡ ( V, W ) = 0 if and only if V and W have nontrivialintersection and d ∡ ( V, W ) = 1 if and only if they are orthogonal to each other. Forother cases, d ∡ ( V, W ) falls between and .If v , . . . , v r are vectors and w = w ∧ · · · ∧ w s the wedge product of an orthonor-mal basis of W , then(11) k v ∧ · · · ∧ v r ∧ w k = k π W ⊥ ( v ) ∧ · · · ∧ π W ⊥ ( v r ) k . In particular, if ( v , . . . , v r ) is an orthonormal basis of V , then(12) d ∡ ( V, W ) = k π W ⊥ ( v ) ∧ · · · ∧ π W ⊥ ( v r ) k . If f : V → W is a linear map between euclidean spaces of same dimension, thenthe determinant of its matrix expressed in orthonormal bases up to a sign does notdepend on the choice of the bases. Moreover, we have | det( f ) | = k f ( v ) ∧ · · · ∧ f ( v r ) k where ( v , . . . , v r ) is an orthonormal basis of V . Together with (12) this gives yetanother definition of d ∡ ( V, W ) if dim( V ) + dim( W ) = n ,(13) d ∡ ( V, W ) = | det( π W ⊥ | V ) | , where π W ⊥ | V : V → W ⊥ denotes the restriction of π W ⊥ to V .The natural action of the orthogonal group O( n ) on the Grassmannian preserves d ∡ , i.e. ∀ g ∈ O( n ) , d ∡ ( gV, gW ) = d ∡ ( V, W ) . Consequently if dim V + dim W = n then(14) d ∡ ( V ⊥ , W ⊥ ) = d ∡ ( V, W ) , because in this case we can always send V to W ⊥ (hence W to V ⊥ ) by an elementof O( n ) .Moreover, when we have several subspaces, V , V , . . . , V q of R n , we define d ∡ ( V , . . . , V q ) = k v ∧ · · · ∧ v q k where for each i = 1 , . . . , q , v i is the wedge product of the elements of an orthonor-mal basis of V i . For example, if x , . . . , x n ∈ R n are unit vectors, then d ∡ ( R x , . . . , R x n ) = | det( x , . . . x n ) | . Obviously, d ∡ ( V , . . . , V q ) is symmetric in the variables V , . . . , V q . Below aresome other elementary properties of d ∡ . WEIKUN HE
Lemma 13. If U, V, W are linear subspaces of R n , then (15) d ∡ ( U, V, W ) = d ∡ ( U + V, W ) d ∡ ( U, V ) . Consequently, if V , . . . , V q are also linear subspaces, then (16) d ∡ ( V , . . . , V q ) = d ∡ ( V , V ) d ∡ ( V , V + V ) · · · d ∡ ( V q , V + · · · + V q − ); (17) d ∡ ( V + · · · + V q , W ) ≥ d ∡ ( V , W ) d ∡ ( V , V + W ) · · · d ∡ ( V q , V + · · · + V q − + W ) . Proof.
If the sum U + V is not a direct sum, then d ∡ ( U, V, W ) = 0 and d ∡ ( U, V ) =0 . Otherwise, let u and v be wedge products of orthonormal bases of U and V respectively. Then u ∧ v / k u ∧ v k is the wedge product of an orthonormal basis of U + V . Then (15) follows immediately from the definition.The estimates (16) can be obtained by a simple induction. The inequality(17) follows from (16) since, by (16), the right hand side of (17) is equal to d ∡ ( V , . . . , V q , W ) which, by (16) again, is equal to d ∡ ( V , . . . , V q ) d ∡ ( V + · · · + V q , W ) . (cid:3) Lemma 14.
Let q ≥ . Let V , . . . , V q be linear subspaces of R n . If z ∈ V + · · · + V q then (18) k z k d ∡ ( V , . . . , V q ) ≤ k π V ( z ) k + k π V ( z ) k + · · · + k π V q ( z ) k Proof.
We will proceed by induction. Let q = 2 . Obviously, there is nothing toprove if V + V is not a direct sum. Moreover, without loss of generality, we canassume that R n = V + V . Hence also R n = V ⊥ + V ⊥ . Write z = z + z with z ∈ V ⊥ and z ∈ V ⊥ . Then by (12), k π V ( z ) k = k π V ( z ) k = k z k d ∡ ( V ⊥ , R z ) ≥ k z k d ∡ ( V ⊥ , V ⊥ ) = k z k d ∡ ( V , V ) . Similarly, k π V ( z ) k ≥ k z k d ∡ ( V , V ) . We get the lemma for q = 2 using thetriangular inequality.Now, suppose the lemma is true for some q ≥ . Let us show the lemma for q + 1 . Let V ′ q = V q + V q +1 and z ′ = π V ′ q ( z ) . The induction hypothesis applied to z and ( V , . . . , V q − , V ′ q ) gives k z k d ∡ ( V , . . . , V q − , V q + V q +1 ) ≤ k π V ( z ) k + · · · + k π V q − ( z ) k + k z ′ k . The q = 2 case applied to z ′ and ( V q , V q +1 ) gives k z ′ k d ∡ ( V q , V q +1 ) ≤ k π V q ( z ′ ) k + k π V q +1 ( z ′ ) k = k π V q ( z ) k + k π V q +1 ( z ) k . Recall that d ∡ ( V , . . . , V q +1 ) = d ∡ ( V , . . . , V q − , V q + V q +1 ) d ∡ ( V q , V q +1 ) . We ob-tain the desired estimate by multiplying the first inequality by d ∡ ( V q , V q +1 ) andcombining it with the second. (cid:3) Lemma 15. If R n is a direct sum of V , . . . , V q then for any bounded subset A ⊂ R n , (19) N δ ( A ) ≪ n d ∡ ( V , . . . , V q ) − n q Y i =1 N δ ( π V i ( A )) . Proof.
Suppose for each i ∈ { , . . . , q } , π V i ( A ) is covered by the balls x ( δ ) i , x i ∈ X i ⊂ V i . For each ( x i ) i ∈ X × · · · × X q , there is a unique x ∈ R n such that ∀ i, π V i ( x ) = x i . By Lemma 14, we have π − V ( x ( δ )1 ) ∩ · · · ∩ π − V q ( x ( δ ) q ) ⊂ x ( δ ′ ) , where δ ′ = d ∡ ( V , . . . , V q ) − qδ . So A is covered by the balls centered at such x .Hence N δ ′ ( A ) ≤ | X | · · · | X q | . We then conclude by using the scale change estimate(8). (cid:3) RTHOGONAL PROJECTIONS OF DISCRETIZED SETS 9
Lemma 16.
Let
V, W, U be linear subspaces of R n , with U ⊂ W . We have (20) d ∡ ( V, U + W ⊥ ) = d ∡ ( V, W ⊥ ) d ∡ ( π W ( V ) , U ) . Proof.
Both sides of (20) vanish if the dimension of V ′ = π W ( V ) is smaller thanthat of V . So we can assume that dim V ′ = dim V = r . Let ( v , . . . , v r ) be anorthonormal basis of V . Then ( π W ( v ) , . . . , π W ( v r )) is a basis of V ′ . Moreover,by (11), we have k π W ( v ) ∧ · · · ∧ π W ( v r ) k = d ∡ ( V, W ⊥ ) and k π W ( v ) ∧ · · · ∧ π W ( v r ) ∧ u k = d ∡ ( V, U, W ⊥ ) , where u is the wedge product an orthonormal basis of U . The desired equality (20)follows from the fact d ∡ ( V ′ , U ) = k π W ( v ) ∧ · · · ∧ π W ( v r ) ∧ u kk π W ( v ) ∧ · · · ∧ π W ( v r ) k and Lemma 13 applied to V, U, W ⊥ : d ∡ ( V, U, W ⊥ ) = d ∡ ( U, W ⊥ ) d ∡ ( V, U + W ⊥ ) = d ∡ ( V, U + W ⊥ ) . (cid:3) Lemma 17.
Let
V, W be linear subspaces of R n . If V ′ = π W ( V ) , then for all x ∈ W , (21) d ∡ ( V, W ⊥ ) k π V ′ ( x ) k ≤ k π V ( x ) k ≤ k π V ′ ( x ) k . Proof.
Since V ′ = π W ( V ) , we have V ′⊥ ∩ W ⊂ V ⊥ . Hence we can write x = y + z with y = π V ′ ( x ) ∈ V ′ and z ∈ V ′⊥ ∩ W ⊂ V ⊥ . Then π V ( x ) = π V ( y ) . This givesthe second inequality in (21).It is clear that V and V ′ have different dimensions if and only if V and W ⊥ havenontrivial intersection, which is equivalent to d ∡ ( V, W ⊥ ) = 0 . In this case, the firstinequality in the lemma holds.Thus we can assume dim V = dim V ′ . By (12), k π V ( y ) k = d ∡ ( R y, V ⊥ ) k y k . Weknow that d ∡ ( R y, V ⊥ ) ≥ d ∡ ( V ′ , V ⊥ ) = d ∡ ( V, V ′⊥ ) by the fact that R y ⊂ V ′ and (14). Observe that V ′⊥ = V ′⊥ ∩ W + W ⊥ . Also, d ∡ ( V, V ′⊥ ∩ W + W ⊥ ) = d ∡ ( V, W ⊥ ) d ∡ ( V ′ , V ′⊥ ∩ W ) = d ∡ ( V, W ⊥ ) , by Lemma 16 applied to V, W and U = V ′⊥ ∩ W . Hence k π V ( y ) k ≥ d ∡ ( V, W ⊥ ) k y k ,which proves the first inequality in (21). (cid:3) Lemma 18.
Let
V, W be linear subspaces of R n such that d ∡ ( V, W ⊥ ) > . Write V ′ = π W ( V ) . For any bounded subset A ⊂ W , (22) N δ ( π V ′ ( A )) ≪ n d ∡ ( V, W ⊥ ) − n N δ ( π V ( A )) . In particular, if moreover dim V = dim W , then for any bounded subset A ⊂ W , (23) N δ ( A ) ≪ n d ∡ ( V, W ⊥ ) − n N δ ( π V ( A )) . Proof.
Since d ∡ ( V, W ⊥ ) > , π V restricted to W is surjective. Hence we can cover π V ( A ) by the balls π V ( b ) ( δ ) , b ∈ ˜ A for some ˜ A ⊂ W with | ˜ A | = N δ ( π V ( A )) . Then π V ′ ( A ) is covered by the balls π V ′ ( b ) ( δ ′ ) , b ∈ ˜ A with δ ′ = d ∡ ( V, W ⊥ ) − δ . Indeed, ∀ a ∈ A , there is b ∈ ˜ A such that k π V ( a − b ) k ≤ δ . Hence, by (21), k π V ′ ( a − b ) k ≤ δ ′ .Thus N δ ′ ( π V ′ ( A )) ≤ N δ ( π V ( A )) , which yields (22) using (8). (cid:3) Intersections.
Here we collect two useful lemmata about intersections andunions of intersections.The first one is about intersections of large subsets. Let A be a Borel set in R n .Let Θ be an index set equipped with a probability measure µ and for each θ ∈ Θ ,we have a Borel subset A θ of A . We need appropriate measurability, namely, themap ( x, θ ) A θ ( x ) is required to be measurable. Lemma 19.
In the situation described above, if there is K ≥ such that ∀ θ ∈ Θ , λ ( A θ ) ≥ λ ( A ) /K , then for any positive integer q > , µ ⊗ q ( (cid:8) ( θ , . . . , θ q ) | λ ( A θ ∩ · · · ∩ A θ q ) ≥ λ ( A )2 K q (cid:9) ) ≥ K q . Proof.
By Fubini’s theorem and then Jensen’s inequality, Z λ ( A θ ∩ · · · ∩ A θ q ) d µ ⊗ q ( θ , . . . , θ q )= Z A Z A θ ( x ) · · · A θq ( x ) d µ ⊗ q ( θ , . . . , θ q ) d λ ( x )= λ ( A ) Z A (cid:0)Z A θ ( x ) d µ ( θ ) (cid:1) q d λ ( x ) λ ( A ) ≥ λ ( A ) (cid:0)Z A Z A θ ( x ) d µ ( θ ) d λ ( x ) λ ( A ) (cid:1) q = λ ( A ) (cid:0)Z λ ( A θ ) λ ( A ) d µ ( θ ) (cid:1) q ≥ λ ( A ) K q . The lemma follows. (cid:3)
The next lemma is about small probability events happening simultaneously. Let ( E, µ ) be a probability space. Suppose we have a collection of subsets ( E i ) i ∈{ ,...,N } of E . We will think E i as events with small probability and we want to estimatethe probability such that a lot of them happen together. Here "a lot" is relativelyto weights we give to the events. Let ( a i ) i ∈{ ,...,N } be non-negative real numberssuch that P Ni =1 a i = 1 . For I ⊂ { , . . . , N } , write a I = P i ∈ I a i . The followinglemma is an easy consequence of Markov’s inequality. Lemma 20.
With the notation above, we have, for any a > , µ (cid:0) [ I | a I ≥ a (cid:0)\ i ∈ I E i (cid:1)(cid:1) ≤ a − max i ∈{ ,...,N } µ ( E i ) . Proof.
Consider the Bernoulli random variables X i = E i for i = 1 , . . . , N so that µ ( E i ) = E (cid:2) X i (cid:3) and µ (cid:0) [ I | a I ≥ a \ i ∈ I E i (cid:1) = P (cid:2) N X i =1 a i X i ≥ a (cid:3) . Then it follows from Markov’s inequality that P (cid:2) N X i =1 a i X i ≥ a (cid:3) ≤ a − E (cid:2) N X i =1 a i X i (cid:3) ≤ a − max i ∈{ ,...,N } E (cid:2) X i (cid:3) . This finishes the proof. (cid:3)
RTHOGONAL PROJECTIONS OF DISCRETIZED SETS 11
Additive combinatorial tools.
Let
A, B, C be bounded subsets of R n . Welook at them at scale δ > . We will use several well-known results from additivecombinatorics in our metric entropic setting. We shall use some usual notationfrom additive combinatorics : A + B = { a + b | a ∈ A, b ∈ B } ,A − B = { a − b | a ∈ A, b ∈ B } , and for integer k ≥ , kA denotes the k -fold sumset A + · · · + A . Lemma 21 (Ruzsa triangular inequality) . We have N δ ( B ) N δ ( A − C ) ≪ n N δ ( A − B ) N δ ( B − C ) . Lemma 22 (Plünnecke-Ruzsa inequality) . For all K ≥ , if N δ ( A + B ) ≤ K N δ ( B ) then for all natural number k and l , N δ ( kA − lA ) ≪ n K k + l N δ ( B ) . Both lemmata above can be obtained by approximating R n by the lattice δ. Z d and then using its discrete counterpart (see for example [25]) as a black box. Moreprecisely for a subset A ⊂ R n , we define ˜ A = (cid:8) a ∈ δ · Z n | A ∩ a ( nδ ) = ∅ (cid:9) . Then A ⊂ ˜ A ( nδ ) and ˜ A ⊂ A ( nδ ) . These inclusions behave nicely under addition andsubtraction.Before stating the Balog-Szemerédi-Gowers theorem in the discretized setting letus recall some basic facts about energy in the discrete setting. Let ϕ : X → Y be amap between discrete sets and A a finite subset of X , define the ϕ -energy of A tobe ω ( ϕ, A ) = X y ∈ Y | A ∩ ϕ − ( y ) | . In other words, it is the square of the l -norm of the push-forward of the countingmeasure on A under ϕ or the number of collisions of the map ϕ | A : ω ( ϕ, A ) = k ϕ ∗ A k = { ( a , a ) ∈ A × A : ϕ ( a ) = ϕ ( a ) } . For example, the usual additive energy between two subsets A and B in an abeliangroup G is ω (+ , A × B ) where + : G × G → G denotes the group law of G .When nothing is known about ϕ , ω ( ϕ, A ) can be as small as | A | (when ϕ isinjective) and as large as | A | (when ϕ is constant on A ). If the image of A by ϕ is small then the energy is large by the Cauchy-Schwarz inequality :(24) ω ( ϕ, A ) ≥ | A | | ϕ ( A ) | . The converse is not true. Nevertheless, we have a partial converse.
Lemma 23.
Suppose there are
K, M > such that ω ( ϕ, A ) ≥ MK | A | and for all y ∈ Y , | A ∩ ϕ − ( y ) | ≤ M . Then there exists A ′ ⊂ A such that | A ′ | ≥ K | A | and | ϕ ( A ′ ) | ≤ KM | A | .Proof. The idea is to trim off small fibers. We consider Y ′ = (cid:8) y ∈ Y | | A ∩ ϕ − ( y ) | ≥ M K (cid:9) and let A ′ = ϕ − ( Y ′ ) . By the definition Y ′ , we have | A | ≥ X y ∈ Y ′ | A ∩ ϕ − ( y ) | ≥ M K | Y ′ | . Hence | ϕ ( A ′ ) | ≤ KM | A | .From the definition of the energy, ω ( ϕ, A ) ≤ M K X y / ∈ Y | A ∩ ϕ − ( y ) | + M X y ∈ Y ′ | A ∩ ϕ − ( y ) |≤ M K | A | + M | A ′ | . It follows that | A ′ | ≥ K | A | . (cid:3) What the Balog-Szemerédi-Gowers theorem roughly says is that if ϕ is a grouplaw (or has some injectivity property similar to a group law) and A is a Cartesianproduct then the conclusion of A ′ in the conclusion of the lemma can be chosen tobe a Cartesian product.For discretized sets we have an analogous notion of energy. Let ϕ : X → Y be amap between metric spaces and A a bounded subset of X . We define the ϕ -energyof A at scale δ as ω δ ( ϕ, A ) = N δ (cid:0) { ( a, a ′ ) ∈ A × A | d ( ϕ ( a ) , ϕ ( a ′ )) ≤ δ } (cid:1) . Here we adhere to the convention that the distance on any Cartesian product X × Y of metric spaces is such that d (cid:0) ( x, y ) , ( x ′ , y ′ ) (cid:1) = d ( x, x ′ ) + d ( y, y ′ ) , for all pairs ( x, y ) , ( x ′ , y ′ ) ∈ X × Y .The analogue of inequality (24) is true. Namely, if A is a bounded subset of R n and ϕ is defined on R n then(25) ω δ ( ϕ, A ) ≫ n N δ ( A ) N δ ( ϕ ( A )) . We also remark that if ψ : A → R n is K -Lipschitz with K ≥ and ϕ : R n → Y is an another map, then it follows from (9) that(26) ω δ ( ϕ, ψA ) ≪ n K n ω δ ( ϕ ◦ ψ, A ) . We will need the following additive version of the Balog-Szemerédi-Gowers the-orem which gives a nice criterion for the additive energy between two sets to belarge. See for example [24, Theorem 6.10] where it is proved in a much broadercontext.
Theorem 24 (Balog-Szemerédi-Gowers theorem) . Let K ≥ be a parameter. Let A and B be bounded subsets of R n . If ω δ (+ , A × B ) ≥ K N δ ( A ) N δ ( B ) , then there exists A ′ ⊂ A and B ′ ⊂ B such that N δ ( A ′ ) ≫ n K − O (1) N δ ( A ) , N δ ( B ′ ) ≫ n K − O (1) N δ ( B ) and N δ ( A ′ + B ′ ) ≪ n K O (1) N δ ( A ) N δ ( B ) . Technical lemmata
In this section, we show the deduction of Theorem 1 from Theorem 6 and collectseveral other lemmata which are needed in the next section. Since they are mostlyabout technical details, it is advisable to skip their proofs for a first reading. Inthis section, implied constants in Landau notations O ( f ) and Vinogradov notations f ≪ g may depend on the dimension n and the parameter κ . Every statement istrue only for δ > sufficiently small and by sufficiently small we mean smaller that RTHOGONAL PROJECTIONS OF DISCRETIZED SETS 13 a constant depending on all other parameters (e.g. n , m , α , κ and ǫ ) but not on A nor on µ . Typically, if C = O (1) then C ≤ δ − ǫ .3.1. Proof of Theorem 1 admitting Theorem 6.
We deduce Theorem 1 fromTheorem 6.
Proposition 25.
Assume that < m < n , < α < n , κ > and ǫ > areparameters that make Theorem 6 true. Let A be a subset of R n contained in theunit ball. Let µ be a probability measure on Gr( R n , m ) . Assume that µ satisfies (3) and A satisfies N δ ( A ) ≥ δ − α + ǫ ; ∀ ρ ≥ δ, ∀ x ∈ R n , N δ ( A ∩ B ( x, ρ )) ≤ δ − ǫ ρ κ N δ ( A ) . Then (27) µ (cid:0) E ( A, ǫ (cid:1) ≤ δ ǫ . The idea is the following. A first application of Theorem 6 gives a subset A ′ ⊂ A with µ ( E ( A ′ , ǫ )) ≤ δ ǫ . Either A ′ is large enough in which case we are done or wecan cut A ′ out of A and apply Theorem 6 again. This will give us another subset A ′ . Then we iterate until the union of these A ′ s is large enough. Proof.
Let N ≥ be an integer. Suppose we have already constructed A , . . . , A N such that A ( δ ) i are pairwise disjoint and µ ( E ( A i , ǫ )) ≤ δ ǫ for every i = 1 , . . . , N .Either we have(28) N δ (cid:0) A \ N [ i =1 A (2 δ ) i (cid:1) ≤ δ ǫ N δ ( A ) , in which case we stop, or the set A \ S Ni =1 A (2 δ ) i satisfies both (1) and (2). In thelatter case Theorem 6 gives us A N +1 ⊂ A \ S Ni =1 A (2 δ ) i with µ ( E ( A N +1 , ǫ )) ≤ δ ǫ .By construction, A ( δ ) N +1 is disjoint with any of A ( δ ) i , i = 1 , . . . , N .When this procedure ends write A = S Ni =1 A i . Then (28) says N δ ( A \ A (2 δ )0 ) ≤ δ ǫ N δ ( A ) . Moreover, by the disjointness of A ( δ )1 , . . . , A ( δ ) N , we have N δ ( A ) = N X i =1 N δ ( A i ) . Set a i = N δ ( A i ) N δ ( A ) . We claim that E ( A, ǫ ⊂ [ I \ i ∈ I E ( A i , ǫ ) , where the index set I runs over subsets of { , . . . , N } with P i ∈ I a i ≥ δ ǫ . Thedesired upper bound (27) then follows immediately from Lemma 20.We now proceed to show the claim. Let V ∈ E ( A, ǫ ) . By definition, there exists A ′ ⊂ A with N δ ( A ′ ) ≥ δ ǫ N δ ( A ) and N δ ( π V ( A ′ )) ≤ δ − mn α − ǫ . Consider the indexset I defined as I = { i ∈ { , . . . , N } | N δ ( A ′ (2 δ ) ∩ A i ) ≥ δ ǫ N δ ( A i ) } . We have, by Lemma 12 and (10), N δ ( A ′ ) − N δ ( A \ A (2 δ )0 ) ≤ n X i =1 N δ ( A ′ ∩ A (2 δ ) i ) ≪ X i ∈ I N δ ( A i ) + X i/ ∈ I N δ ( A ′ (2 δ ) ∩ A i ) ≪ X i ∈ I a i N δ ( A ) + δ ǫ N δ ( A ) Hence P i ∈ I a i ≥ δ ǫ . On the other hand, for all i ∈ I , since N δ ( π V ( A ′ (2 δ ) ∩ A i )) ≤ N δ ( π V ( A ′ ) (2 δ ) ) ≪ N δ ( π V ( A ′ )) , we have N δ ( π V ( A ′ (2 δ ) ∩ A i )) ≤ δ − mn α − ǫ . Hence V ∈ E ( A i , ǫ ) for all i ∈ I . This finishes the proof of the claim. (cid:3) Action of linear transformations.
Clearly, all the assumptions and theconclusion of Theorem 6 are invariant under the action of the orthogonal group O( n ) . The next proposition states that the action of a δ − ǫ -bi-Lipschitz lineartransformation only affects them by a factor of δ O ( ǫ ) . Here, while f ∈ GL( R n ) acts on R n in the usual way, it acts on the Grassmannian by multiplication by f ⊥ := ( f − ) ∗ or equivalently, f ⊥ V = ( f V ⊥ ) ⊥ for all V ∈ Gr( R n , m ) . Lemma 26.
Let < m < n be dimensions. Let ǫ > . Let f ∈ GL( R n ) with k f k + k f − k ≤ δ − ǫ . Let A be a bounded subset of R n and µ a probability measureon Gr( R n , m ) .(i) For each of the conditions (1) – (3) of Theorem 1, if it holds for A and µ with the parameters α , κ and ǫ then it also holds for the image set f A andthe image measure f ⊥∗ µ with the parameters α , κ and O ( ǫ ) in the place of ǫ .(ii) For all V ∈ Gr( R n , m ) , N δ ( π f ⊥ V ( f A )) ≤ δ − O ( ǫ ) N δ ( π V ( A )) .(iii) We have µ ( E ( A, ǫ )) ≤ ( f ⊥∗ µ ) (cid:0) E ( f A, O ( ǫ )) (cid:1) . In particular, if the conclusionof Theorem 6 holds for f A and f ⊥∗ µ with some ǫ ′ > in the place of ǫ thenit holds for A and µ with ǫ = ǫ ′ O (1) .(iv) For all V ∈ Gr( R n , m ) and all x ∈ f ⊥ V , N δ (cid:0) f A ∩ π − f ⊥ V ( x ( δ ) ) (cid:1) ≤ δ − O ( ǫ ) max y ∈ V N δ (cid:0) A ∩ π − V ( y ( δ ) ) (cid:1) . Proof.
The statement about the conditions (1) and (2) follows immediately fromthe inequality (9). As for the condition (3), it suffices to prove that for all W ∈ Gr( R n , n − m ) and all ρ ≥ δ ,(29) f ⊥∗ µ (cid:0) V ∡ ( f ⊥ W, ρ ) (cid:1) ≤ µ (cid:0) V ∡ ( W, δ − O ( ǫ ) ρ ) (cid:1) . From the Cartan decomposition of f , we see easily that ∀ r = 1 , . . . , n , k V r f ⊥ k + k ( V r f ⊥ ) − k ≤ δ − O ( ǫ ) . For V ∈ Gr( R n , m ) , let v be the wedge product of anorthonormal basis of V and w that of W . We have d ∡ ( f ⊥ V, f ⊥ W ) = k ( V n f ⊥ )( v ∧ w ) kk ( V m f ⊥ ) v k k ( V n − m f ⊥ ) w k≥ k ( V n f ⊥ ) − k − k v ∧ w kk V m f ⊥ k k V n − m f ⊥ k≥ δ O ( ǫ ) d ∡ ( V, W ) . Hence f ⊥ V ∈ V ∡ ( f ⊥ W, ρ ) implies V ∈ V ∡ ( W, δ − O ( ǫ ) ρ ) , which establishes (29). RTHOGONAL PROJECTIONS OF DISCRETIZED SETS 15
For the second statement, observe that there is a finite set ˜ A of cardinality | ˜ A | = N δ ( π V ( A )) such that A ⊂ ˜ A + V ⊥ + B (0 , δ ) . Applying f and then π f ⊥ V on both sides, we obtain π f ⊥ V ( f A ) ⊂ π f ⊥ V ( f ˜ A ) + B (0 , δ − ǫ ) . This proves that N δ − ǫ ( π f ⊥ V ( f A )) ≤ N δ ( π V ( A )) . We conclude by the scale changeestimate (8).For the next statement, it suffices to prove that f ⊥ V ∈ E ( f A, O ( ǫ )) when-ever V ∈ E ( A, ǫ ) . Indeed, let V ∈ E ( A, ǫ ) . Then there exists A ′ ⊂ A suchthat N δ ( A ′ ) ≥ δ ǫ N δ ( A ) and N δ ( π V ( A ′ )) < δ − mn α − ǫ . On the one hand, by (9),we have N δ ( f A ′ ) ≥ δ O ( ǫ ) N δ ( f A ) . On the other hand, from (ii) it follows that N δ ( π f ⊥ V ( f A ′ )) ≤ δ − mn α − O ( ǫ ) . Hence f ⊥ V ∈ E ( f A, O ( ǫ )) .For the last statement, it suffices to prove that for any x ∈ f ⊥ V , there exists y ∈ V such that(30) f A ∩ π − f ⊥ V ( x ( δ ) ) ⊂ f (cid:0) A ∩ π − V ( y ( δ − ǫ ) ) (cid:1) . Indeed, if a ∈ A satisfies π f ⊥ V ( f ( a )) ∈ x ( δ ) , then f ( a ) ∈ x + f V ⊥ + B (0 , δ ) . Applying f − and then π V on both sides, we obtain π V ( a ) ∈ π V ( f − ( x )) + B (0 , δ − ǫ ) . This proves (30) with y = π V ( f − ( x )) . (cid:3) Non-concentration property for projections.
Let A a subset of R n asin Theorem 6. We want to understand whether a projection of A still satisfiessome similar regularity property as A does. More precisely we want to find V ∈ Gr( R n , m ) and a large subset A ′ of A such that ∀ ρ ≥ δ, ∀ x ∈ V, N δ ( π V ( A ′ ) ∩ x ( ρ ) ) ≤ ρ κ δ − mn α − ǫ ′ , for some κ > proportional to κ and some ǫ ′ > proportional to ǫ .In the special case where m divides n , we have the following result. We will onlyneed this non-concentration result in this special case, although it might be true ina more general context. Lemma 27.
Let n = qm with q ≥ . For any parameters < α < n , κ > and ǫ > , the following is true for δ > sufficiently small. If A is a subset of R n contained in the unit ball and µ is a probability measure on Gr( R n , m ) satisfyingthe assumptions (1) – (3) for the parameters α , κ and ǫ , then µ ( E ( A ) \ E reg ( A )) ≤ δ ǫ , where E reg ( A ) denotes the set of all V ∈ E ( A ) such that ∃ A ′ ⊂ A with N δ ( A ′ ) ≥ δ ǫ N δ ( A ) and N δ ( π V ( A ′ )) ≤ δ − mn α − ǫ and (31) ∀ ρ ≥ δ, ∀ x ∈ V, N δ ( π V ( A ′ ) ∩ x ( ρ ) ) ≤ ρ κ q δ − mn α − ǫ . The idea of the proof is the following. When V ∈ E ( A ) , there is a large subset A ′ with small projection to V . We then remove small fibers of the projection π V : A ′ → V to get A ′′ . Any large subset in of π V ( A ′′ ) will have large preimage by π V . Thus if V / ∈ E reg ( A ) then there will be a cylinder with axis V ⊥ and radius ρ in which A is very dense. If there are a lot of such V we can then intersect thesecylinders to get a ball of radius ρ q which will contradict the non-concentrationproperty (2) of A . Proof.
For conciseness, write κ = κ q . We claim that if V ∈ E ( A ) \ E reg ( A ) thenthere exists x ∈ V and ρ ≥ δ such that(32) N δ (cid:0) A ∩ π − V ( x ( ρ ) ) (cid:1) ≥ ρ κ δ − ǫ N δ ( A ) . Indeed, let V ∈ E ( A ) \ E reg ( A ) . Then from the definition (6) there exists A ′ ⊂ A with N δ ( A ′ ) ≥ δ ǫ N δ ( A ) and N δ ( π V ( A ′ )) ≤ δ − αq − ǫ . Now we remove small fibers ofthe map π V restricted to A ′ . Consider the set B = (cid:8) y ∈ V | N δ (cid:0) A ′ ∩ π − V ( y ( δ ) ) (cid:1) ≥ δ αq +3 ǫ N δ ( A ) (cid:9) and A ′′ = A ′ ∩ π − V ( B ( δ ) ) . We have, for all y ∈ V , N δ (cid:0) ( A ′ \ A ′′ ) ∩ π − V ( y ( δ ) ) (cid:1) ≤ δ αq +3 ǫ N δ ( A ) . for otherwise y would belong to B and the intersection ( A ′ \ A ′′ ) ∩ π − V ( y ( δ ) ) wouldbe empty. Consequently, N δ ( A ′ \ A ′′ ) ≤ N δ ( π V ( A ′ )) max y ∈ V N δ (cid:0) ( A ′ \ A ′′ ) ∩ π − V ( y ( δ ) ) (cid:1) ≤ δ ǫ N δ ( A ) . It follows that N δ ( A ′′ ) ≥ δ ǫ N δ ( A ) . But V / ∈ E reg ( A ) , the non-concentration prop-erty (31) fails for π V ( A ′′ ) : there exists x ∈ V and ρ ≥ δ such that(33) N δ (cid:0) π V ( A ′′ ) ∩ x ( ρ ) (cid:1) ≥ ρ κ δ − αq − ǫ . Let ˜ B be a maximal δ -separated subset of π V ( A ′′ ) ∩ x ( ρ ) . From (7) and (33), wehave | ˜ B | ≫ ρ κ δ − αq − ǫ . Moreover for all y ∈ ˜ B , by the definition of A ′′ , y ∈ B ( δ ) ,hence N δ (cid:0) A ′ ∩ π − V ( y (2 δ ) ) (cid:1) ≥ δ αq +3 ǫ N δ ( A ) . Since ˜ B is δ -separated, all the balls y (2 δ ) with center y ∈ ˜ B are δ -away from each other. Consequently, N δ (cid:0) A ′ ∩ π − V ( x ( ρ +2 δ ) ) (cid:1) ≥ X y ∈ ˜ B N δ (cid:0) A ′ ∩ π − V ( y (2 δ ) ) (cid:1) ≥ | ˜ B | δ αq +3 ǫ N δ ( A ) ≥ ( ρ + 2 δ ) κ δ − ǫ N δ ( A ) . This finishes the proof of the claim.To obtain a contradiction, suppose that µ ( E ( A ) \ E reg ( A )) ≥ δ ǫ . Note thatthe radius ρ in the claim depends on V . Nevertheless, from (32) we know that itranges from δ to δ ǫκ . For the argument below, we want (32) to hold for a lot of V ∈ E ( A ) \ E reg ( A ) with some radius ρ ≥ δ independent of V . Indeed, by a simplepigeonhole argument , we can find a subset D ⊂ E ( A ) \ E reg ( A ) and a radius ρ ≥ δ such that µ ( D ) ≥ δ ǫ and for all V ∈ D , there exists x ∈ V such that N δ ( A ∩ π − V ( x ( ρ ) )) ≥ ρ κ δ − ǫ N δ ( A ) and hence, by Lemma 12, λ ( A ( δ ) ∩ π − V ( x ( ρ ) )) ≥ ρ κ δ − ǫ λ ( A ( δ ) ) . Let V , . . . , V q be random elements of Gr( R n , m ) independently distributed ac-cording to µ . On the one hand, from Lemma 19 applied to the restriction of µ to D , it follows that with probability at least ( ρ κ δ − ǫ ) q µ ( D ) q ≥ ρ qκ δ − qǫ , thereexists x ∈ V , . . . , x q ∈ V q such that(34) λ (cid:0) A ( δ ) ∩ π − V ( x ( ρ )1 ) ∩ · · · ∩ π − V q ( x ( ρ ) q ) (cid:1) ≥ ρ qκ δ − qǫ λ ( A ( δ ) ) . On the other hand, from (16) and (3), it follows that with probability at least − ( q − δ − ǫ ρ κq , we have(35) d ∡ ( V , . . . , V q ) ≥ ρ q − q . Arrange different ρ into intervals of the form [ δ − k , δ − k − ] , where ≤ k ≪ − log( ǫ ) . RTHOGONAL PROJECTIONS OF DISCRETIZED SETS 17
Now with our choice of κ , we have − ( q − δ − ǫ ρ κq + ρ qκ δ − qǫ > . Thismeans that for some ( V , . . . , V q ) , both (34) and (35) hold. By Lemma 14, thereexists x ∈ R n such that π − V ( x ( ρ )1 ) ∩ · · · ∩ π − V q ( x ( ρ ) q ) ⊂ x ( ρ ′ ) with ρ ′ = qρ d ∡ ( V , . . . , V q ) − ≤ qρ q . Then the non-concentration property (2) of A implies that λ (cid:0) A ( δ ) ∩ x ( ρ ′ ) (cid:1) ≪ δ − ǫ ρ κq λ ( A ( δ ) ) . Combining this with (34) yields ρ qκ δ − qǫ ≪ δ − ǫ ρ κq , which is impossible with our choice of κ . (cid:3) Non-concentration property for slices.
We shall also consider slices of A ,i.e. intersection of A with a δ -neighborhood of a affine subspace. When n = qm ,we have similar non-concentration results for ( n − m ) -dimensional slices of A . Lemma 28.
Let n = qm with q ≥ a positive integer. Let < α < n , κ > and ǫ > be parameters. If the statement in Theorem 6 fails for the set A , then thereis a ( n − m ) -dimensional affine subspace y + W and a subset B ⊂ A ( δ ) ∩ ( y + W ) such that N δ ( B ) ≥ δ − β + O ( ǫ ) and (36) ∀ ρ ≥ δ, ∀ x ∈ W N δ ( B ∩ x ( ρ ) ) ≤ ρ κ q δ − β − O ( ǫ ) , where β = q − q α . Here is an outline of the proof. The negation of Theorem 6 to A implies thatthere is a large subset A q ⊂ A occupying a large portion of the Cartesian product Q qj =1 π V j ( A q ) of its projections to q subspaces in nearly orthogonal position. Then,because of Lemma 27, the first factor π V ( A q ) can be chosen to have the non-concentration property. This in turn will imply the non-concentration property ofthe projection of A q to V + · · · + V q − . Then it would suffice to find a slice whoseprojection to V + · · · + V q − is nearly as large as that of A q , which can be easilydone given the negation of Theorem 6. Proof.
Suppose the statement in Theorem 6 fails for the set A ⊂ R n . This means µ ( E ( A ′ )) > δ ǫ for any subset A ′ ⊂ A . In particular, E reg ( A ) is non-empty byLemma 27. Let V ∈ E reg ( A ) . There exists A ⊂ A with N δ ( A ) ≥ δ − α +3 ǫ and(37) ∀ ρ ≥ δ, ∀ x ∈ V , N δ ( π V ( A ) ∩ x ( ρ ) ) ≤ ρ κ q δ − q α − ǫ . Let ǫ = ǫκ . We construct by a simple induction a sequence of subspaces V , . . . , V q and a nested sequence of subsets A ⊃ · · · ⊃ A q satisfying for any j = 2 , . . . , q , d ∡ ( V j , V + · · · + V j − ) ≥ δ ǫ , (38) N δ ( A j ) ≥ δ ǫ N δ ( A j − ) , N δ ( π V j ( A j )) ≤ δ − q α − ǫ . (39)This is possible since at each step, we have by (3), µ (cid:0) E ( A j − ) \ V ∡ ( V + · · · + V j − , δ ǫ ) (cid:1) ≥ δ ǫ − δ ǫ > . From the fact that N δ ( A q ) ≤ N δ ( π V q ( A q )) max y ∈ V q N δ ( A q ∩ π − V q ( y ( δ ) )) , we get some y ⋆ ∈ V q such that(40) N δ ( A q ∩ π − V q ( y ( δ ) ⋆ )) ≥ δ − q − q α + O ( ǫ ) . After a translation, we can suppose y ⋆ = 0 . We write V = V + · · · + V q − and W = V ⊥ q and set B = A q ∩ W ( δ ) and B = π W ( B ) . We have N δ ( B ) ≥ δ − β + O ( ǫ ) from (40) and the fact that B ⊂ B ( δ ) .It remains to show the non-concentration property (36) for B . Let ρ ≥ δ and x ∈ W . From (38), d ∡ ( V, W ⊥ ) = d ∡ ( V, V q ) ≥ δ O ( ǫ ) . Hence, by (23) in Lemma 18, N δ ( B ∩ x ( ρ ) ) ≤ δ − O ( ǫ ) N δ ( π V ( B ) ∩ x ( ρ )0 ) where x = π V ( x ) . Moreover B ⊂ A ( δ ) q , hence, by (10), N δ ( π V ( B ) ∩ x ( ρ )0 ) ≤ δ − ǫ N δ ( π V ( A q ) ∩ x (2 ρ )0 ) . Then Lemma 15 applied to the set π V ( A q ) ∩ x (2 ρ )0 in V = L q − j =1 V j together with(38) yield N δ ( π V ( A q ) ∩ x (2 ρ )0 ) ≤ δ − O ( ǫ ) N δ ( π V ( A q ) ∩ x (2 ρ )1 ) q − Y j =2 N δ ( π V j ( A q )) where x = π V ( x ) . The required non-concentration property (36) then followsfrom (37) and (39). (cid:3) Without the non-concentration property.
As illustrated by the examplein the introduction, the non-concentration condition (2) on A is crucial to have again ǫ > in the conclusion. Without this condition, we can still expect N δ ( π V ( A )) to be close to N δ ( A ) mn for generic V ∈ Gr( R n , m ) . This is the subject of the nextproposition. Proposition 29.
Given < m ≤ n , < α < n and κ > , there exists C > such that for all < ǫ < κC , the following is true for all δ > sufficiently small.Let A ⊂ R n be a subset contained in the unit ball and µ a probability measure on Gr( R n , m ) . Assume that (41) N δ ( A ) ≥ δ − α − Cǫ . Further assume the non-concentration property (3) for µ if m < n . Then µ ( E ( A )) ≤ δ ǫ . When m divides n , this follows almost immediately from Lemma 15. Then thetask is to reduce to this special case. Since it shares the same set of ideas as theproof of Theorem 6, the proof below will only be outlined and more details can befound in the next section. Proof.
For < m ≤ n , denote by P ( n, m ) the statement we want to show. Notethat for all n ≥ , P ( n, n ) is trivially true. We will proceed by an induction similarto that in the proof of Theorem 6. It suffices to show the following two types ofinductive steps. Let < m ≤ n and q, r > be integers.(i) If mq ≤ n , then P ( n, qm ) implies P ( n, m ) .(ii) If n = q ( n − m ) + r with < r ≤ n − m , then P ( n, r ) and P ( n − r, m ) imply P ( n, m ) .Using the same argument in Proposition 25, we see that in order to show P ( n, m ) ,it suffices to show µ ( E ( A ′ )) ≤ δ ǫ for some subset A ′ ⊂ A . In other words, if theconclusion of P ( n, m ) fails for the set A then for any subset A ′ ⊂ A , µ ( E ( A ′ )) ≥ δ ǫ . RTHOGONAL PROJECTIONS OF DISCRETIZED SETS 19
Proof of (i) . Let V , . . . , V q be random elements of Gr( R n , m ) independentlydistributed according to µ . Write V = V + · · · + V q . When qm < n , we know byLemma 33 that P (cid:2) dim( V ) = qm (cid:3) ≥ − ( q − δ κ − ǫ , and the distribution of V conditional to the event dim( V ) = qm has the corre-sponding non-concentration property. By P ( n, qm ) , we know that for any C ′ > ,if the constant C in (41) is large enough (depending on C ′ ) then the probabilitythat there exists A ′ ⊂ A satisfying N δ ( A ′ ) ≥ δ C ′ ǫ N δ ( A ) and N δ ( π V ( A ′ )) ≤ δ − qmn α − C ′ ǫ is at most δ C ′ ǫ + ( q − δ κ − ǫ .Suppose that P ( n, m ) fails for A . Then by a simple induction we show that withprobability at least δ O ( ǫ ) , we have d ∡ ( V , . . . , V q ) ≥ δ O ( ǫ ) and there exists A q ⊂ A such that N δ ( A q ) ≥ δ O ( ǫ ) N δ ( A ) and ∀ j = 1 , . . . , q, N δ ( π V j ( A q )) ≤ δ − mn α − ǫ and hence, by (19) applied to π V ( A ) in V = L qj =1 V j , N δ ( π V ( A )) ≤ δ − qmn α − O ( ǫ ) . We obtain a contradiction if C ′ were chosen to be larger than any of the implicitconstants in the Landau notations appearing above. Proof of (ii), Case 1 . Assume firstly that A contains large slice of dimension n − r . More precisely, assume that there exists W ∈ Gr( R n , n − r ) and x ∈ R n suchthat N δ (cid:0) A ∩ ( x + W ( δ ) ) (cid:1) ≥ δ − n − rn α − C ′ ǫ where C ′ is the constant given by P ( n − r, m ) applied to < m ≤ n − r , n − rn α and κ .Without loss of generality, we can assume that x = 0 and that B = π W ( A ∩ W ( δ ) ) is contained in A . Lemma 31 tells us that we can apply P ( n − r, m ) to B ⊂ W with the image measure of µ by π W . Then we can conclude using Lemma 32. Proof of (ii), Case 2 . Otherwise A does not contain any large slice of dimension n − r :(42) ∀ x ∈ R n , ∀ W ∈ Gr( R n , n − r ) , N δ (cid:0) A ∩ ( x + W ( δ ) ) (cid:1) ≤ δ − n − rn α − O ( ǫ ) . Let V , . . . , V q be random elements of Gr( R n , m ) independently distributed ac-cording to µ . Write V = V ∩ · · · ∩ V q . By (14) and Lemma 33 applied to V ⊥ + · · · + V ⊥ q = V ⊥ , P (cid:2) dim( V ) = r (cid:3) ≥ − ( q − δ κ − ǫ and that the distribution of V conditional to the event dim( V ) = r has a non-concentration property. By P ( n, r ) , we know that for any C ′ > , if the constant C in (41) is large enough (depending on C ′ ) then the probability that there exists A ′ ⊂ A satisfying N δ ( A ′ ) ≥ δ C ′ ǫ N δ ( A ) and N δ ( π V ( A ′ )) ≤ δ − rn α − C ′ ǫ is at most δ C ′ ǫ + ( q − δ κ − ǫ .Suppose that P ( n, m ) fails for A . Again by an induction we show that withprobability at least δ O ( ǫ ) , we have d ∡ ( V ⊥ , . . . , V ⊥ q ) ≥ δ O ( ǫ ) and there exists A q ⊂ A such that N δ ( A q ) ≥ δ O ( ǫ ) N δ ( A ) and ∀ j = 1 , . . . , q, N δ ( π V j ( A q )) ≤ δ − mn α − ǫ Together with (42), this implies by Proposition 34 that there exists A ′ ⊂ A q suchthat N δ ( A ′ ) ≥ δ O ( ǫ ) N δ ( A ) and N δ ( π V ( A ′ )) ≤ δ − rn α − O ( ǫ ) . Again we obtain a contradiction if C ′ is large compared to any of the impliedconstants in the previous Landau notations. (cid:3) Proof of the main result
In this section, we prove Theorem 6 and thus Theorem 1. This is done byproving first the base case where n = 2 m (Propsoition 7) and then the inductionsteps (Propositions 9-11). Note that on account of Proposition 25, for a given pair ( n, m ) , if Theorem 6 is true for these dimensions then so is Theorem 1. Therefore,when we use Theorem 6 as induction hypothesis, the conclusion is µ ( E ( A )) ≤ δ ǫ while when we prove by contradiction by saying that A is a counterexample forTheorem 6, we are assuming µ ( E ( A ′ )) > δ ǫ for all subsets A ′ of A .Like in the previous section, all implied constants in Landau and Vinogradovnotations in this section may depend on n and κ . Again, every statement in thissection is true only for δ > smaller than a constant depending on n , m , α , κ and ǫ .4.1. Half dimensional projections.
For the special case n = 2 m , we followmainly the proof in [2] (which deals with the case m = 1 ) while using a techniquein the proof of Proposition 2 in Bourgain-Glibichuk [4]. The main idea, as explainedin the introduction, is to reduce to the case where A is a Cartesian product X ⋆ × X ⋆ with the help of Balog-Szemerédi-Gowers theorem and then apply a sum-productestimate. Proof of Proposition 7.
Suppose Theorem 6 fails for the subset A ⊂ R n and theprobability measure µ on Gr( R n , m ) with n = 2 m . We will get a contradictionwhen ǫ is small enough. By Lemma 27, there is a subspace V and a subset A ⊂ A with the following properties: N δ ( A ) ≥ δ − α +3 ǫ , N δ ( π V ( A )) ≤ δ − α − ǫ and(43) ∀ ρ ≥ δ, ∀ x ∈ V , N δ ( π V ( A ) ∩ x ( ρ ) ) ≤ ρ κ δ − α − O ( ǫ ) . Let ǫ = ǫκ . Then µ (cid:0) E ( A ) \V ∡ ( V , δ ǫ ) (cid:1) ≥ δ ǫ − δ ǫ > by the non-concentrationproperty (3) of µ . Let V ∈ E ( A ) \ V ∡ ( V , δ ǫ ) with A such that(44) N δ ( A ) ≥ δ − α +4 ǫ and N δ ( π V j ( A )) ≤ δ − α − ǫ , j = 1 , . Observe that since d ∡ ( V , V ) ≥ δ O ( ǫ ) there exists a δ − O ( ǫ ) -bi-Lipschitz map f ∈ GL( R n ) satifying (recalling the notation introduced in Subsection 3.2) f ⊥ V = V and f ⊥ V = V ⊥ . Applying Lemma 26 to f , we see that we can assume withoutloss of generality that V = V ⊥ .Put X = π V ( A ) and Y = π V ( A ) . We have, N δ ( A ) ≪ N δ ( X ) N δ ( Y ) and thistogether with the inequalities (44) implies N δ ( X ) , N δ ( Y ) ≥ δ − α + O ( ǫ ) . Write D = E ( A ) \ ( V ∡ ( V , δ ǫ ) ∪V ∡ ( V , δ ǫ )) . We have, by (3), µ ( D ) ≥ δ ǫ − δ ǫ ≥ δ ǫ . Let V ∈ D . By (13) and (14), we have | det( π V | V ) | = d ∡ ( V , V ⊥ ) = d ∡ ( V , V ) ≥ δ O ( ǫ ) . The same is true for π V | V . Then it follows easily from the Cartan decompositionthat(45) k π − V | V k ≤ δ − O ( ǫ ) and k π − V | V k ≤ δ − O ( ǫ ) . RTHOGONAL PROJECTIONS OF DISCRETIZED SETS 21
Since V ∈ E ( A ) , there is a subset A V ⊂ A such that N δ ( A V ) ≥ δ − α + O ( ǫ ) and N δ ( π V ( A V )) ≤ δ − α − ǫ . It follows from (25) that ω δ ( π V , X + Y ) ≥ ω δ ( π V , A V ) ≥ δ − α + O ( ǫ ) . By (45), the map R n = V ⊕ V → V × V defined by v + v ( π V ( v ) , π V ( v )) is δ − O ( ǫ ) -bi-Lipschitz. Hence, by (26), we can bound from below the additive energybetween π V X and π V Y , ω δ (+ , π V X × π V Y ) ≥ δ − α + O ( ǫ ) ≥ δ O ( ǫ ) N δ ( π V X ) N δ ( π V Y ) . That is why we can apply the Balog-Szemerédi-Gowers theorem (Theorem 24)to get subsets X V ⊂ X and Y V ⊂ Y such that(46) N δ ( X V ) , N δ ( Y V ) ≥ δ − α + O ( ǫ ) and(47) N δ ( π V X V + π V Y V ) ≤ δ − α − O ( ǫ ) . Applying π − V | V to the set in the last inequality and using (9), we obtain(48) N δ ( X V + ϕ V Y V ) ≤ δ − α − O ( ǫ ) , where ϕ V : V → V is ϕ V = π − V | V ◦ π V | V . Note that from (45), ϕ V is δ − O ( ǫ ) -bi-Lipschitz.Let us apply Lemma 19 to the collection of subsets X ( δ ) V × Y ( δ ) V ⊂ X ( δ ) × Y ( δ ) with the restriction of µ to D . We obtain V ⋆ ∈ D , X ⋆ := X V ⋆ and Y ⋆ := Y V ⋆ suchthat λ ( X ( δ ) ⋆ ∩ X ( δ ) V ) λ ( Y ( δ ) ⋆ ∩ Y ( δ ) V ) ≥ δ n − α + O ( ǫ ) whenever V ∈ D ′ , where D ′ is a subset of D with(49) µ ( D ′ ) ≥ δ O ( ǫ ) µ ( D ) ≥ δ O ( ǫ ) . By Ruzsa’s triangular inequality (Lemma 21), (48) implies, for all V ∈ D ′ N δ ( X V − X ( δ ) ⋆ ∩ X ( δ ) V ) ≪ N δ ( X V − X V ) ≤ δ − α − O ( ǫ ) . For the same reason N δ ( X ⋆ − X ( δ ) ⋆ ∩ X ( δ ) V ) ≤ δ − α − O ( ǫ ) . Then by Ruzsa’s triangularinequality again, we have(50) N δ ( X ⋆ − X V ) ≤ δ − α − O ( ǫ ) . Similarly, N δ ( Y ⋆ − Y V ) ≤ δ − α − O ( ǫ ) , which implies with (9),(51) N δ ( ϕ V Y ⋆ − ϕ V Y V ) ≤ δ − α − O ( ǫ ) . Moreover, (48) specified to V = V ∗ with (9) applied to ϕ V ϕ − ⋆ gives(52) N δ ( ϕ V ϕ − ⋆ X ⋆ + ϕ V Y ⋆ ) ≤ δ − α − O ( ǫ ) , where ϕ ⋆ := ϕ V ⋆ .Now successive use of Ruzsa’s triangular inequality (recalling (50), (48), (51)and (52)) yields that for all V ∈ D ′ ,(53) N δ ( X ⋆ − ϕ V ϕ − ⋆ X ⋆ ) ≤ δ − α − O ( ǫ ) . Moreover, by the Plünnecke-Ruzsa inequality (Lemma 22),(54) N δ ( X ⋆ + X ⋆ ) ≤ δ − α − O ( ǫ ) . Consider the set of endomorphisms A = {− ϕ V ϕ − ⋆ ∈ End( V ) | V ∈ D ′ } . Weclaim that the assumptions of Theorem 8 are satisfied for A and X ⋆ with σ = α ,and κ replaced by κ and ǫ replaced by O ( ǫ ) . Therefore, when ǫ is small enough,(53) and (54) contradict Theorem 8. Our claim about the assumptions (i), (iv) and (vi) are clear from what precedes.The assumption (v) follows from (43) and (46) because for any ρ ≥ δ , N δ ( X ⋆ ) ≤ N ρ ( X ⋆ ) max x ∈ V N δ ( X ⋆ ∩ x ( ρ ) ) . In the case of m = 1 , the assumption (iii) is trivially true and the assumption(ii) follows immediately from (3) the fact that d ∡ is a distance on Gr( R , and thefact that the map Gr( R , \ V ∡ ( V , δ ǫ ) → R , V ϕ V ϕ − ⋆ is δ − O ( ǫ ) -bi-Lipschitz.Finally, to prove (ii) and (iii) in the case where m ≥ we use Lemma 30 below.For any f ∈ End( V ) , pick an arbitrary nonzero vector v ∈ V and apply Lemma 30to v = ϕ − ⋆ ( v ) and W = R f ( v ) . This gives the existence of a subspace W ′ ∈ Gr( R n , m ) such that d ∡ ( V, W ′ ) ≤ δ − O ( ǫ ) d ( ϕ V ϕ − ⋆ , f ) . Hence by (3), for any ρ ≥ δ , µ (cid:0) { V ∈ D ′ | − ϕ V ϕ − ⋆ ∈ B ( f, ρ ) } (cid:1) ≤ δ − O ( ǫ ) ρ κ . Observe that µ ( D ′ ) ≤ N ρ ( A ) max f ∈ End( V ) µ (cid:0) { V ∈ D ′ | − ϕ V ϕ − ⋆ ∈ B ( f, ρ ) } (cid:1) . Together with (49), this gives the assumption (ii), namely, N ρ ( A ) ≥ δ O ( ǫ ) ρ − κ . Moreover, for any nonzero proper linear subspace W ∈ V , take w ∈ W somevector with k w k = 1 and consider ρ = sup V ∈D ′ d ( − ϕ V ϕ − ⋆ ( w ) , W ) . By Lemma 30 applied to v = ϕ − ⋆ ( w ) which has norm ≤ δ − O ( ǫ ) , we have D ′ ⊂V ∡ ( W ′ , δ − O ( ǫ ) ρ ) for some W ′ ∈ Gr( R n , m ) . In view of (49) and (3), we have δ O ( ǫ ) ≤ δ − O ( ǫ ) ρ κ . Hence ρ ≥ δ O ( ǫ ) , which establishes (iii). (cid:3) Lemma 30.
We use the notations in the proof above. For any nonzero vector v ∈ V and any proper linear subspace W ⊂ V , there is W ′ ∈ Gr( R n , m ) such thatfor all V ∈ Gr( R n , m ) , (55) d ∡ ( V, W ′ ) ≤ k v k − d ( ϕ V ( v ) , W ) . Proof.
Without loss of generality, we can assume that dim( W ) = m − . For any V ∈ Gr( R n , m ) , any v ∈ V and any w ∈ W , by (12), we have d ∡ ( V ⊥ , R ( v − w )) = k π V ( v − w ) kk v − w k . Note that k v − w k ≥ k v k since v ⊥ w and k π V ( v − w ) k ≤ k ϕ V ( v ) − w k since π V ( ϕ V ( v ) − w ) = π V ( v − w ) . Hence d ∡ ( V ⊥ , R ( v − w )) ≤ k ϕ V ( v ) − w kk v k . As w can be any vector in W , we obtain d ∡ ( V ⊥ , R v + W ) ≤ k v k − d ( ϕ V ( v ) , W ) . We conclude by setting W ′ = ( R v + W ) ⊥ ∈ Gr( R n , m ) and using (14). (cid:3) RTHOGONAL PROJECTIONS OF DISCRETIZED SETS 23
Projection of a slice.
If the set A contains a relatively large slice of di-mension < n ′ < n (a subset B = A ( δ ) ∩ ( y + W ) with dim( W ) = n ′ and N δ ( B ) ≍ δ − n ′ n α ) and if it has a correct non-concentration property then we canapply the induction hypothesis to B − y inside W . Instead of projecting to V distributed according to µ , we project to V ′ = π W ( V ) . The first lemma belowshows that V ′ is not concentrated and the next one shows the relationship between V ′ being in E ( B ) ∩ Gr(
W, m ) and V being in E ( B ) . Using this idea we proveProposition 9. Lemma 31.
Let < m < n ′ < n be integers and κ, ǫ > be parameters. Let W ∈ Gr( R n , n ′ ) and V be a random element of Gr( R n , m ) having the followingnon-concentration property, (56) ∀ ρ ≥ δ, ∀ U ∈ Gr( R n , n − m ) , P (cid:2) d ∡ ( V, U ) ≤ ρ (cid:3) ≤ δ − ǫ ρ κ . Set V ′ = π W ( V ) . Then with probability at least − δ κ − ǫ , dim( V ′ ) = m . Conditionalto this event the distribution of V ′ is a probability measure ν on Gr(
W, m ) . Itsatisfies ∀ ρ ≥ δ, ∀ U ∈ Gr(
W, n ′ − m ) , ν ( V ∡ ( U, ρ )) ≤ δ − ǫ ρ κ . Proof.
We know that dim( V ′ ) = m if and only if d ∡ ( V, W ⊥ ) > . The first partfollows immediately from (56) specified to ρ = δ .Let us show the non-concentration property for ν . Let U be a ( n ′ − m ) -dimen-sional subspace of W . By Lemma 16, we have d ∡ ( V, U + W ⊥ ) ≤ d ∡ ( V ′ , U ) . Hencefor all ρ ≥ δ , by (56) P (cid:2) d ∡ ( V ′ , U ) ≤ ρ (cid:3) ≤ P (cid:2) d ∡ ( V, U + W ⊥ ) ≤ ρ (cid:3) ≤ δ − ǫ ρ κ and hence ν ( V ∡ ( U, ρ )) ≤ δ − ǫ ρ κ − δ κ − ǫ ≤ δ − ǫ ρ κ . (cid:3) Lemma 32.
Let < m ≤ n ′ < n be integers. Let < α < n and ǫ > beparameters. Let B ⊂ W be a bounded subset in a n ′ -dimensional linear subspace W ⊂ R n . Then π W (cid:0) E ( B, ǫ ) \ V ∡ ( W ⊥ , δ ǫ ) (cid:1) ⊂ E ( B, O ( ǫ )) ∩ Gr(
W, m ) . Proof.
Let V ∈ E ( B, ǫ ) \ V ∡ ( W ⊥ , δ ǫ ) . Then there exists B ′ ⊂ B such that N δ ( B ′ ) ≥ δ ǫ N δ ( B ) and N δ ( π V ( B ′ )) ≤ δ − mn α − ǫ . Denote by V ′ the projection π W ( V ) . It follows from Lemma 18 that N δ ( π V ′ ( B ′ )) ≤ d ∡ ( V, W ⊥ ) − O (1) N δ ( π V ( B ′ )) ≤ δ − mn α − O ( ǫ ) . That is why V ′ ∈ E ( B, O ( ǫ )) ∩ Gr(
W, m ) . (cid:3) Proof of Proposition 9.
Let n = qm and suppose that Theorem 6 holds for n ′ =( q − m and m . Let A and µ be as in Theorem 6 but for which the conclusionfails. By Lemma 28, there is an n ′ -dimensional affine subspace y + W and a subset B ⊂ A ( δ ) ∩ ( y + W ) such that N δ ( B ) ≥ δ − β + O ( ǫ ) and ∀ ρ ≥ δ, ∀ x ∈ W, N δ ( B ∩ x ( ρ ) ) ≤ ρ κ q δ − β − O ( ǫ ) where β = q − q α . Without loss of generality, we can assume y = 0 and B ⊂ A .Let V be a random element of Gr( R n , m ) distributed according to µ . Define ν be as in Lemma 31. By the lemma, we can apply the induction hypothesis(Theorem 6 combined with Proposition 25) to B ⊂ W with the probability measure ν on Gr(
W, m ) . We obtain a constant ǫ ′ > depending only on n ′ , β and κ suchthat when ǫ ≤ ǫ ′ , ν (cid:0) E ( B, ǫ ′ ) ∩ Gr(
W, m ) (cid:1) ≤ δ ǫ ′ . Set ǫ = ǫκ . By Lemma 32, we have µ (cid:0) E ( B, ǫ ) \ V ∡ ( W ⊥ , δ ǫ ) (cid:1) ≤ ν (cid:0) E ( B, O ( ǫ )) ∩ Gr(
W, m ) (cid:1) . When ǫ ≤ ǫ ′ O (1) , the last two inequalities together with (3) yield µ ( E ( B, ǫ )) ≤ µ (cid:0) E ( B, ǫ ) \ V ∡ ( W ⊥ , δ ǫ ) (cid:1) + µ ( V ∡ ( W ⊥ , δ ǫ )) ≤ δ ǫ ′ + δ ǫ ≤ δ ǫ , which finishes the proof of Proposition 9. (cid:3) Projection to a sum of subspaces.
In the situation where m < n , weconsider the sum V = V + · · · + V q where q is a positive integer such that qm < n and V , . . . , V q are m -dimensional subspaces. Using the inequality (19), the size ofthe projection to V can be bounded in terms of the sizes of the projections to each V j . In the next lemma, we prove that if V j are independently randomly distributedaccording to a measure with an appropriate non-concentration property then thedistribution of their sum V has a non-concentration property as well. This allowsus to apply the induction hypothesis with the dimensions n and m ′ = qm . Thisidea leads to the proof of Proposition 10. Lemma 33.
Let n, m, q, r be positive integers such that qm + r = n . Let <ǫ < κ be parameters. Let V , . . . V q be independent random elements of Gr( R n , m ) satisfying ∀ j = 1 , . . . , q , ∀ ρ ≥ δ, ∀ W ∈ Gr( R n , n − m ) P (cid:2) d ∡ ( V j , W ) ≤ ρ (cid:3) ≤ δ − ǫ ρ κ . Then with probability at least − ( q − δ κ − ǫ , we have (57) dim( V + · · · + V q ) = qm. Then the probability measure µ ′ on Gr( R n , qm ) defined as the distribution of V + · · · + V q conditional to the event (57) satisfies the non-concentration property ∀ ρ ≥ δ, ∀ W ∈ Gr( R n , r ) , µ ′ ( V ∡ ( W, ρ )) ≤ δ − O ( ǫ ) ρ κq . Proof.
Let V , . . . , V q be as in the statement. By their independence, for every j = 2 , . . . , q , P (cid:2) d ∡ ( V j , V + · · · + V j − ) ≤ δ (cid:3) ≤ δ κ − ǫ . Hence, on account of (16), with probability at least − ( q − δ κ − ǫ , we have d ∡ ( V , . . . , V q ) ≥ δ ( q − > and hence V + · · · + V q is a direct sum.Let ρ ≥ δ and W ∈ Gr( R n , r ) . By (17), we know that if d ∡ ( V + · · · + V q , W ) ≤ ρ then for some j = 1 , . . . , q , d ∡ ( V j , V + · · · + V j − + W ) ≤ ρ q , which happens with probability at most δ − ǫ ρ κq . Therefore, P (cid:2) d ∡ ( V + · · · + V q , W ) ≤ ρ (cid:3) ≤ qδ − ǫ ρ κq . Hence µ ′ ( V ∡ ( W, ρ )) ≤ qδ − ǫ ρ κq − ( q − δ κ − ǫ ≤ δ − O ( ǫ ) ρ κq . (cid:3) RTHOGONAL PROJECTIONS OF DISCRETIZED SETS 25
Proof of Proposition 10.
Let n, m, q, r be positive integers such that qm + r = n .Suppose Theorem 6 is true for the dimensions n and m ′ = qm but it fails for thedimensions n and m with parameters < α < n , κ > and ǫ > . Let A and µ be a counterexample, i.e. A and µ satisfy (1)–(3) but µ ( E ( A ′ )) > δ ǫ for all subsets A ′ ⊂ A . We will get a contradiction when ǫ is smaller than a constant dependingonly on n , α and κ .Let V , . . . V q be random elements of Gr( R n , m ) independently distributed ac-cording to µ . Write V = V + · · · + V q and let µ ′ be the distribution of V con-tional to the event dim( V ) = qm as in Lemma 33. It is a probability measureon Gr( R n , qm ) satisfying a non-concentration property, according to Lemma 33.Thus, we can apply the induction hypothesis (Theorem 6 combined with Proposi-tion 25) with dimensions n and m ′ = qm to the set A and the measure µ ′ . It gives ǫ ′ = ǫ ′ ( n, α, κ ) > such that for all ǫ ≤ ǫ ′ , the probability that there exists A ′ ⊂ A satisfying N δ ( A ′ ) ≥ δ ǫ ′ N δ ( A ) and N δ ( π V ( A ′ )) ≤ δ − qmn α − ǫ ′ is at most δ ǫ ′ + ( q − δ κ − ǫ .The rest of the proof consist of proving a lower bound for the same probability.First, V ∈ E ( A ) with probability at least δ ǫ . When this happens, there is A ⊂ A with N δ ( A ) ≥ δ ǫ N δ ( A ) and N δ ( π V ( A )) ≤ δ − mn α − ǫ . Write ǫ = ǫκ . Thenconditional to any choice of V , we have V ∈ E ( A ) \ V ∡ ( V , δ ǫ ) with probabilityat least δ ǫ . When this happens, there is A ⊂ A with N δ ( A ) ≥ δ ǫ N δ ( A ) and N δ ( π V ( A )) ≤ δ − mn α − ǫ . Then conditional to any choice of V and V , theprobability that V ∈ E ( A ) \ V ∡ ( V + V , δ ǫ ) is at least δ ǫ . We continue thisconstruction until we get A q .To summarize, we have with probability at least δ (2 q − ǫ , d ∡ ( V , . . . , V q ) ≥ δ O ( ǫ ) and there exists a subset A q ⊂ A satisfying N δ ( A q ) ≥ δ qǫ N δ ( A ) and for every j = 1 , . . . , q , N δ ( π V j ( A q )) ≤ δ − mn α − ǫ and hence, by Lemma 15 applied to π V ( A ) in V = ⊕ qj =1 V j , N δ ( π V ( A q )) ≤ δ − qmn α − O ( ǫ ) . This leads to a contradiction when ǫ ≤ ǫ ′ O (1) . (cid:3) Projection to intersection of subspaces I: a discrete model.
When theprojections of a set A to subspaces V , . . . , V q are all small, we would like to say thatits projection to the intersection V = V ∩ · · · ∩ V q is small as well. This is not true.A typical example is A = ( R e ⊕ R e ) ∪ R e where ( e , e , e ) is the standard basisin R . While its projections to R e ⊕ R e and to R e ⊕ R e are both small (havedimension in a -dimensional space), its projection to R e is full dimensional. Inthis example, A contains a large slice orthogonal to V . This happens to be themajor obstruction. Proposition 34.
Let n, m, q, r be positive integers such that n = q ( n − m ) + r . Forany < α < n and ǫ > , the following is true for sufficiently small δ > . Let A ⊂ R n and V , . . . , V q ∈ Gr( R n , m ) . Write V = V ∩ · · · ∩ V q . Assume that(i) d ∡ ( V ⊥ , . . . , V ⊥ q ) ≥ δ ǫ ;(ii) δ − α + ǫ ≤ N δ ( A ) ≤ δ − α − ǫ ;(iii) For every j = 1 , . . . , q , N δ ( π V j ( A )) ≤ δ − mn α − ǫ ;(iv) For all y ∈ V , N δ (cid:0) A ∩ π − V ( y ( δ ) ) (cid:1) ≤ δ − n − rn α − ǫ . Then there exists A ′ ⊂ A such that N δ ( A ′ ) ≥ δ O ( ǫ ) N δ ( A ) and N δ ( π V ( A ′ )) ≤ δ − rn α − O ( ǫ ) . This proposition is deduced from the following discrete analogue. Let n, m, q, r be as in Proposition 34. For I ⊂ { , . . . , n } , we write ̟ I : Z n → Z I to denote thediscrete projection ( z i ) i ∈{ ,...,n } ( z i ) i ∈ I . Consider I = { n − r + 1 , . . . , n } andfor j = 1 , . . . , qI j = { , . . . , n } \ { ( j − n − m ) + 1 , . . . , j ( n − m ) } . Proposition 35.
We use the notations above. For any parameter K ≥ and anyfinite subset Z ⊂ Z n . One of the following statements is true.(i) There exists j ∈ { , . . . , q } such that | ̟ I j ( Z ) | ≥ K | Z | mn .(ii) There exists y ∈ Z I such that | Z ∩ ̟ − I ( y ) | ≥ K | Z | n − rn .(iii) There exists Z ′ ⊂ Z such that | Z ′ | ≥ K q +1 | Z | and | ̟ I ( Z ′ ) | ≤ K q | Z | rn . One of the ingredients is a discrete projection inequality due to Bollobás andThomason [1] known as the uniform cover theorem. Let P ( { , . . . , n } ) denote theset of subsets of { , . . . , n } . Recall that a multiset of subsets of { , . . . , n } is acollection of elements of P ( { , . . . , n } ) which can have repeats. Giving such amultiset is equivalent to giving a map from P ( { , . . . , n } ) to N . Following Bollobás-Thomason, we say a multiset C is k -uniform cover of { , . . . , n } if each element i ∈ { , . . . , n } belongs to exactly k members of C . For exemple, with I j definedabove, ( I \ I , . . . , I q \ I ) is a ( q − -uniform cover of { , . . . , n } \ I . Theorem 36 (Uniform Cover theorem, Bollobás-Thomason [1]) . Let Z be a finitesubset of Z n . Let C be an k -uniform cover of { , . . . , n } . Then we have | Z | k ≤ Y I ∈C | ̟ I ( Z ) | . This is a generalisation of an isoperimetric inequality due to Loomis and Whit-ney [16]. For example, if we consider projections onto all canonical m -dimensionalsubspaces. There is always one which has at least the expected size: there exists I ⊂ { , . . . , n } such that | I | = m and | ̟ I ( Z ) | ≥ | Z | m/n . Although the Loomis-Whitney inequality is already sufficient for the proof of Proposition 35, we will workat a slightly greater generality (the lemma below), since it requires no extra effort. Lemma 37.
Let I ⊂ { , . . . , n } . Let Z be a finite subset of Z n and C a k -uniformcover of { , . . . , n } \ I with q elements. Then | Z | q − k ≤ ω ( ̟ I , Z ) q − k Y I ∈C | ̟ I ∪ I ( Z ) | . This lemma is a refinement of the uniform cover theorem. Indeed, for I = ∅ ,we have ω ( ̟ I , Z ) = | Z | and we recover the uniform cover theorem. Proof.
For all I ∈ C , we have | ̟ I ∪ I ( Z ) | = X y ∈ ̟ I ( Z ) | ̟ I ( Z ∩ ̟ − I ( y )) | . Hence, by Hölder’s inequality, X y ∈ ̟ I ( Z ) Y I ∈C | ̟ I ( Z ∩ ̟ − I ( y )) | q ≤ Y I ∈C (cid:0)X y | ̟ I ( Z ∩ ̟ − I ( y )) | (cid:1) q = Y I ∈C | ̟ I ∪ I ( Z ) | q . RTHOGONAL PROJECTIONS OF DISCRETIZED SETS 27
For each y ∈ ̟ I ( Z ) , we apply the uniform cover theorem (Theorem 36) to the set Z ∩ ̟ − I ( y ) seen as a finite subset of Z { ,...,n }\ I , | Z ∩ ̟ − I ( y ) | kq ≤ Y I ∈C | ̟ I ( Z ∩ ̟ − I ( y )) | q . From the two inequalities above, we get k ̟ I ∗ Z k k kq ≤ Y I ∈C | ̟ I ∪ I ( Z ) | . Finally, Hölder’s inequality implies | Z | = k ̟ I ∗ Z k ≤ k ̟ I ∗ Z k k q − kkq k ̟ I ∗ Z k q − k q − k . We finish the proof by putting the last two inequalities together and recalling that ω ( ̟ I , Z ) = k ̟ I ∗ Z k . (cid:3) Proof of Proposition 35.
We use the notations introduced before Proposition 35.Applying Lemma 37 to the ( q − -uniform cover ( I \ I , . . . , I q \ I ) of { , . . . , n }\ I ,we get | Z | q +1 ≤ ω ( ̟ I , Z ) q Y j =1 | ̟ I j ( Z ) | . If the first statement of Proposition 35 does not hold, we would have ω ( ̟ I , Z ) ≥ K q | Z | n − rn . If the second statement fails as well, we can apply Lemma 23 with M = K | Z | n − rn and K ′ = K q +1 . The third statement follows immediately. (cid:3) Proof of Proposition 34.
Let ( e , . . . , e n ) denote the standard basis of R n . Firstconsider the special case where V ⊥ j is exactly Span( e ( j − n − m )+1 , . . . , e j ( n − m ) ) foreach j = 1 , . . . , q . Then we conclude easily from Proposition 35 by setting K = δ − ǫ and Z = { x ∈ Z n | A ∩ δ · ( x + [0 , n ) = ∅ } . For the general case we consider a map f ∈ GL( R n ) which sends isometrically V to Span( e n − r +1 , . . . , e n ) and V ⊥ j to Span( e ( j − n − m )+1 , . . . , e j ( n − m ) ) for each j = 1 , . . . , q . It is easy to see that k f − k ≤ n and | det( f − ) | = d ∡ ( V ⊥ , . . . , V ⊥ q , V ) = d ∡ ( V ⊥ , . . . , V ⊥ q ) . Therefore f is δ − O ( ǫ ) -bi-Lipschitz.The conclusion for A follows from the special case applied to f A . Indeed,by the inequality (9) and Lemma 26, the hypotheses are satisfied for f A and f ⊥ V , . . . , f ⊥ V q with ǫ replaced by O ( ǫ ) . Moreover, the conclusion for f A and f ⊥ V = f ⊥ V ∩· · ·∩ f ⊥ V q implies that for A and V , again by (9) and Lemma 26. (cid:3) Projection to intersection of subspaces II: concluding proof.
Once wehave Proposition 34, to prove Proposition 11, we can use Proposition 29 and ideasin Subsection 4.2 to rule out the case where A has a very large slice and then applythe arguments in Subsection 4.3 to the dual. Proof of Proposition 11.
Let n, m, q, r be as in Proposition 11. Assume that Theo-rem 6 is true for the dimensions n and m ′ = r and assume that A and µ are coun-terexample to Theorem 6 for the dimensions n and m with parameters < α < n , κ > and ǫ > . We begin by making two remarks. Firstly, we can assume that(58) N δ ( A ) ≤ δ − α − O ( ǫ ) , for otherwise, we could conclude directly by using Proposition 29.Secondly, we can also assume that A does not contain very large slice of codi-mension r . More precisely, we can assume that(59) ∀ W ∈ Gr( R n , n − r ) , ∀ x ∈ R n , N δ (cid:0) A ∩ ( x + W ( δ ) ) (cid:1) ≤ δ − n − rn α − O ( ǫ ) . Indeed, if (59) fails, then put B = π W (cid:0) A ∩ ( x + W ( δ ) ) (cid:1) and we can apply Proposi-tion 29 to B ⊂ W to obtain that E ( B ) ∩ Gr(
W, m ) does not support any measurewith the corresponding non-concentration property in Gr(
W, m ) . We can concludeas in Subsection 4.2 by using Lemma 31 and Lemma 32.From now on assume (58) and (59). Let V , . . . , V q be random elements of Gr( R n , m ) independently distributed according to µ . On account of (14), thenon-concentration property (3) implies similar property for the distribution of V ⊥ ,namely, ∀ ρ ≥ δ, ∀ W ∈ Gr( R n , m ) , P (cid:2) d ∡ ( V ⊥ , W ) ≤ ρ (cid:3) ≤ δ − ǫ ρ κ . From Lemma 33 applied to V ⊥ , . . . , V ⊥ q , we know that with probability at least − ( q − δ κ − ǫ , the intersection V = V ∩ · · · ∩ V q has dimension r . Let µ ′ be thedistribution of V conditional to this event. Then by Lemma 33 and (14), µ ′ hasthe following non-concentration property ∀ ρ ≥ δ, ∀ W ∈ Gr( R n , n − r ) , P (cid:2) d ∡ ( V, W ) ≤ ρ (cid:3) ≤ δ − O ( ǫ ) ρ κq . That is why we can apply the induction hypothesis (Theorem 6 combined withProposition 25) to the set A and the measure µ ′ with n and m ′ = r . We obtain ǫ ′ = ǫ ′ ( n, α, κ ) > such that for all ǫ ≤ ǫ ′ , the probability that there exists A ′ ⊂ A satisfying N δ ( A ′ ) ≥ δ ǫ ′ N δ ( A ) and N δ ( π V ( A ′ )) ≤ δ − rn α − ǫ ′ is at most δ ǫ ′ + ( q − δ κ − ǫ .Now we are going to prove a lower bound for this propability, which will lead to acontradiction. As the conclusion of Theorem 6 fails for A , we have µ ( E ( A ′ )) ≥ δ ǫ forall subsets A ′ ⊂ A . Using a similar construction as in the proof of Proposition 10,we prove that with probability at least δ O ( ǫ ) , we have d ∡ ( V ⊥ , . . . , V ⊥ q ) ≥ δ O ( ǫ ) and there exists A q ⊂ A satisfying N δ ( A q ) ≥ δ O ( ǫ ) N δ ( A ) and for all j = 1 , . . . , q , π V j ( A q ) ≤ δ − mn α − ǫ . Therefore, all the hypotheses of Proposition 34 are satisfied for the set A q with O ( ǫ ) in the place of ǫ . In particular, the assumption (ii) is guaranteed by (1) and(58) and the assumption (iv) is guaranteed by (59). Hence there exists a subset A ′ ⊂ A q such that N δ ( A ′ ) ≥ δ O ( ǫ ) N δ ( A ) and N δ ( π V ( A ′ )) ≤ δ − rn α − O ( ǫ ) . This leads to a contradiction when ǫ ≤ ǫ ′ O (1) . (cid:3) Projection of fractal sets
In this section we derive Theorem 2 from Theorem 1 then Corollary 3 and Corol-lary 5 from Theorem 2.
RTHOGONAL PROJECTIONS OF DISCRETIZED SETS 29
Proof of Theorem 2.
To deduce Theorem 2 from Theorem 1 we need toknow how to discretize a fractal set. The idea is the same as in the proof of [2,Theorem 4]. We include a detailed proof here for the sake of completeness. Butbefore that, let us recall Frostman’s lemma.
Theorem 38 (Frostman’s lemma (see [18, Theorem 8.8])) . Let A be a Borel set of R n . If dim H ( A ) > α then there exists a finite nonzero compactly supported Borelmeasure ν with Supp( ν ) ⊂ A such that ∀ ρ > , ∀ x ∈ R n , ν ( B ( x, ρ )) ≤ ρ α . Proof of Theorem 2.
Let < m < n , < α < n , κ > be parameters. Let ǫ > be times the constant given by Theorem 1 applied to these parameters. Let A and µ be a counterexample for Theorem 2 with these parameters. Without loss ofgenerality we can assume A ⊂ B (0 , .After normalizing µ , we can suppose that it is a probability measure such that ∀ ρ > , ∀ W ∈ Gr( R n , n − m ) , µ ( V ∡ ( W, ρ )) ≪ µ ρ κ . Thus, the non-concentration condition (3) of Theorem 1 is satisfied for sufficientlysmall δ .By Frostman’s lemma, there is a nonzero Radon measure ν compactly supportedon A such that(60) ∀ ρ > , ∀ x ∈ R n , ν ( B ( x, ρ )) ≤ ρ α − ǫ . For any V ∈ Supp( µ ) we have dim H ( π V ( A )) < η where η = mn α + 2 ǫ . By thedefinition of Hausdorff dimension, for any k ≥ , there is a cover π V ( A ) ⊂ [ k ≥ k B V,k of π V ( A ) such that each B V,k is a union of at most kη balls of radius − k in V .Set A V,k = π − V ( B V,k ) for V ∈ Supp( µ ) and k ≥ k . Since the sets A V,k , k ≥ k ,cover A , we have X k ≥ k ν ( A V,k ) ≫ ν . Integrating with respect to d µ ( V ) and using Fubini’s theorem, we obtain X k ≥ k Z ν ( A V,k ) d µ ( V ) ≫ ν . This in turn implies that there exists k ≥ k such that µ ( E ) ≫ ν k − , where E = { V ∈ Gr( R n , m ) | ν ( A V,k ) ≫ ν k − } . Now fix this k and set δ = 2 − k so that N δ ( π V ( A V,k )) ≤ δ − η . Note that as we canchoose k arbitrarily large, we can make δ arbitrarily small.Here we cannot apply Theorem 1 directly to the set A because it might not beregular enough. The idea is to partition A into regular parts. Let Q denotes theset of dyadic cubes in R n of side length δ : Q = (cid:8) x + [0 , δ [ n | x ∈ δ · Z n (cid:9) . Put L = (cid:6) nǫ (cid:7) + 1 . For l = 0 , . . . , L , let A l be the union of all cubes Q ∈ Q suchthat δ ( l +1) ǫ ν ( A ) < ν ( Q ) ≤ δ lǫ ν ( A ) . It is easy to see that A l are disjoint and P Ll =0 ν ( A l ) ≥ (1 − δ ǫ ) ν ( A ) . Moreoverfor any l = 0 , . . . , L and any A ′ ⊂ A l which is also a union of cubes in Q , we have δ ( l +1) ǫ N δ ( A ′ ) ν ( A ) ≪ n ν ( A ′ ) ≪ n δ lǫ N δ ( A ′ ) ν ( A ) Hence, if ν ( A l ) > , then for such A ′ ,(61) δ ǫ ν ( A ′ ) ν ( A l ) ≪ n N δ ( A ′ ) N δ ( A l ) ≪ n δ − ǫ ν ( A ′ ) ν ( A l ) Consider L = { ≤ l ≤ L | ν ( A l ) ≥ δ ǫ } , the set of levels with sufficient mass. Forany l ∈ L , by (60), N δ ( A l ) ≫ δ − α + ǫ ν ( A l ) ≥ δ − α +2 ǫ and from (61) and (60), for any ρ ≥ δ and any x ∈ R n , N δ ( A l ∩ B ( x, ρ )) N δ ( A l ) ≪ n δ − ǫ ν ( B ( x, ρ + nδ )) ν ( A l ) ≪ n δ − ǫ ρ α . In other words, the assumptions of Theorem 1 are satisfied for A l .Now for l ∈ L and V ∈ E , let A V,k,l be the union of Q ∈ Q such that Q ⊂ A l and Q ∩ A V,k = ∅ . From the definition of L and E , we know that for any V ∈ E X l ∈L ν ( A V,k,l ) ≥ ν ( A V,k ) − ( L + 1) δ ǫ ≫ ν k − . Hence there exists l ∈ L such that ν ( A V,k,l ) ν ( A l ) ≫ ν k − . Therefore by setting E l = n V ∈ Gr( R n , m ) | ν ( A V,k,l ) ν ( A l ) ≫ ν k − o , we have E = ∪ l ∈L E l .From the lower bound µ ( E ) ≫ ν k − , we find a certain l ∈ L such that µ ( E l ) ≫ ν,L k − . This contradicts Theorem 1 applied to the set A l and the measure µ . Indeed,recalling the notation (6), Theorem 1 says µ ( E ( A l , ǫ )) ≤ δ ǫ . But we have E l ⊂E ( A l , ǫ ) . Because for all V ∈ E l , we have N δ ( A V,k,l ) ≥ δ ǫ N δ ( A l ) by (61) and also N δ ( π V ( A V,k,l )) ≪ N δ ( π V ( A V,k )) ≤ δ − η . (cid:3) Hausdorff dimension of exceptional set.
In this subsection we deduceCorollary 3 from Theorem 2. First recall the Łojasiewicz inequality which we willneed.
Theorem 39 (Łojasiewicz inequality [15, Théorème 2, page 62]) . Let ( M, d ) be areal analytic manifold endowed with a Riemannian distance d and let f : M → R bea real analytic map. If K is a compact subset of M , then there is C > dependingon K and f such that for all x ∈ K , | f ( x ) | ≥ C min(1 , d ( x, Z )) C where Z = { x ∈ M | f ( x ) = 0 } .Proof of Corollary 3. Recall that we work with a Riemannian metric on the Grass-mannian
Gr( R n , m ) which is invariant under the action of the group O( n ) . Observethat the exceptional set of directions { V ∈ Gr( R n , m ) | dim H ( π V ( A )) ≤ mn α + ǫ } is measurable for the Borel σ -algebra on Gr( R n , m ) . Suppose that the Hausdorffdimension of the exceptional set is larger than m ( n − m ) − κ for some κ .Frostman’s lemma is valid for general compact metric spaces (see [18, Theorem8.17]) . Thus there exists a nonzero Radon measure µ supported on this exceptionalset such that for all ρ > and all V ∈ Gr( R n , m ) , µ ( B ( V, ρ )) ≤ ρ m ( n − m ) − κ . We For our situation, we can simply use a local chart and Frostman’s lemma in R n since a localchart of a Riemannian manifold is necessarily bi-Lipschitz to its image (endowed with the inducedEuclidean distance). RTHOGONAL PROJECTIONS OF DISCRETIZED SETS 31 are going to prove that µ satisfies the non-concentration property forbidden byTheorem 2.We fix W ∈ Gr( R n , m ) and apply the Łojasiewicz inequality to the real analyticfunction d ∡ ( · , W ) : Gr( R n , m ) → R . We conclude that there is a constant C > such that for any < ρ ≤ , V ∡ ( W, ρ ) is contained in the ρ ′ -neighborhood of theSchubert cycle V ∡ ( W, with ρ ′ = ( Cρ ) C . By the O( n ) -invariance, the constant C is in fact uniform for all W ∈ Gr( R n , n − m ) . Since the Schubert cycle V ∡ ( W, is a smooth submanifold, we have N ρ ′ ( V ∡ ( W, ( ρ ′ ) ) ≪ n ρ ′− m ( n − m )+1 . Here again,the estimate is uniform in W thanks to the O( n ) -invariance. Therefore, µ ( V ∡ ( W, ρ )) ≤ N ρ ′ ( V ∡ ( W, ( ρ ′ ) ) sup V ∈ Gr( R n ,m ) µ ( B ( V, ρ ′ )) ≪ n ρ κC . This contradicts Theorem 2 if ǫ is sufficiently small and finishes the proof of Corol-lary 3. (cid:3) Restricted family of projections.
Finally, we deduce Corollary 5 from The-orem 2. We will use the following Remez-type inequality due to A. Brudnyi [5].
Theorem 40 (Brudnyi [5, Theorem 1.2, case (b)]) . Let Ω ⊂ R q and Ω ⊂ R p beconnected open sets in Euclidean spaces. Let f : Ω × Ω → R be a real analyticmap, considered as a family of real analytic functions on Ω depending analyticallyon a parameter varying in Ω . Let K ⊂ Ω and K ⊂ Ω be compact subsets. For x ∈ Ω , write M ( x ) = max y ∈ K | f ( x, y ) | . There exists d ≥ and C > such that for any < ρ < , (62) max x ∈ K N ρ ( { y ∈ K | | f ( x, y ) | < M ( x ) ρ d } ) ≤ Cρ − p . The constant d in (62) depends not only on f but also on K and K . It can beestimated if more is known about the function f . For example, if for any x ∈ Ω , f ( x, · ) : Ω → R is polynomial of degree less than k ∈ N , then d can be taken tobe k . We refer the reader to the introduction in [5] for more details.The function M is lower semicontinuous. Hence by compactness M has a min-imum on K . As a consequence, if we assume that for any x ∈ K , we have M ( x ) > , i.e. f ( x, · ) is not identically zero, then the conclusion (62) can bereformulated as (the constant C becomes larger in this formulation) max x ∈ K N ρ ( { y ∈ K | | f ( x, y ) | ≤ ρ d } ) ≤ Cρ − p . Moreover, by covering with charts, it is easy to see that the same holds if Ω and Ω are connected real analytic manifolds. Proof of Corollary 5.
Let ǫ > as be given by Theorem 2. Consider the realanalytic map f : Gr( n, n − m ) × Ω → R defined by ∀ ( W, t ) ∈ Gr( n, n − m ) × Ω , f ( W, t ) = d ∡ ( V ( t ) , W ) . By the transversality assumption, for any W , the partial function f ( W, · ) is notidentically zero. Hence by Brudnyi’s theorem, there exists d and C > such thatfor all < ρ < ,(63) max W ∈ Gr( n,n − m ) N ρ ( { t ∈ Ω ′ | | f ( W, t ) | ≤ ρ d } ) ≤ Cρ − p . Now assume for a contradiction that the set of exceptional parameters has Hausdorffdimension larger than p − dκ . Then by Frostman’s lemma, there exists a nonzeroBorel measure µ supported on this exceptional set satisfying sup t ∈ R p µ ( B ( t, ρ )) ≤ ρ p − dκ . Then by (63), for any W ∈ Gr( R n , n − m ) , µ ( { t ∈ Ω ′ | d ∡ ( V ( t ) , W ) ≤ ρ d } ) ≤ ρ p − dκ Cρ − p ≤ Cρ dκ In other words the image measure C V ∗ µ has the non-concentration property for-bidden by Theorem 2. This concludes the proof of (5).The moreover part follows from the fact we know the exact value of d in Theo-rem 40 when the map f is polynomial. (cid:3) References [1] B. Bollobás and A. Thomason. Projections of bodies and hereditary properties of hypergraphs.
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Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, Université Paris-Saclay,91405 Orsay, France.Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem91904, Israel.
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