Rigidity, graphs and Hausdorff dimension
N. Chatzikonstantinou, A. Iosevich, S. Mkrtchyan, J. Pakianathan
aa r X i v : . [ m a t h . C A ] A ug RIGIDITY, GRAPHS AND HAUSDORFF DIMENSION
N. CHATZIKONSTANTINOU, A. IOSEVICH, S. MKRTCHYAN AND J. PAKIANATHAN
Abstract.
For a compact set E ⊂ R d and a connected graph G on k + 1 vertices,we define a G -framework to be a collection of k + 1 points in E such that thedistance between a pair of points is specified if the corresponding vertices of G areconnected by an edge. We regard two such frameworks as equivalent if the specifieddistances are the same. We show that in a suitable sense the set of equivalencesof such frameworks naturally embeds in R m where m is the number of “essential”edges of G . We prove that there exists a threshold s k < d such that if the Hausdorffdimension of E is greater than s k , then the m -dimensional Hausdorff measure of theset of equivalences of G -frameworks is positive. The proof relies on combinatorial,topological and analytic considerations. Contents
1. Introduction 22. Definitions and statements of results 32.1. Statements of results 63. Graph distances of subsets of R d G k +1 ,m , E k +1 ) 83.3. A sharp upper bound for the dimension of the distance set 103.4. Bounds on the number of noncongruent realizations 123.5. The proof of the dimensional threshold 123.6. The natural measure ν g on E − gE Date : June 19, 2018.The second and fourth listed authors were partially supported by the NSA Grant H98230-15-1-0319. Introduction
The Falconer distance conjecture ([10]) says that if the Hausdorff dimension of acompact subset of R d , d ≥
2, is greater than d , then the Lebesgue measure of the setof distances determined by pairs of elements of the set is positive. The best currentresults are due to Wolff ([22]) in dimension 2 and Erdogan ([8]) in R d , d ≥
3, whoestablished the d + threshold. In the context of Ahlfors-David regular sets, theFalconer conjecture was established in the plane by Orponen ([20]). These resultsbuild partly on the previous work by Bourgain ([5]), Falconer ([10]) and Mattila(see [19] and the references contained therein). A related conjecture, also pioneeredby Falconer ([10]) and studied extensively by Bourgain, says that if the Hausdorffdimension of E is equal to d , then the upper Minkowski dimension of ∆( E ) is 1.Bourgain proved that if dim H ( E ) = d , E ⊂ R d , d ≥
2, then the upper Minkowskidimension of ∆( E ) is > + ǫ d for some ǫ d > k + 1)-point frameworks with k >
1. Forexample, one can consider triangles inside sufficiently large sets, properly interpreted.This problem has been extensively studied in a variety of contexts by Bennett, Bour-gain, Chan, Furstenberg, Greenleaf, Katznelson, Laba, Pramanik, Weiss, Ziegler,the second and fourth named authors and others (see e.g. [4], [12], [6], [1], [2], [15],[17], [11], [25]). In [3], the authors considered chains, and in [13] necklaces wereinvestigated. More general frameworks were studied in [16].In this paper we show that in a suitable sense, a nontrivial dimensional thresh-old can be found for any finite point framework. What we mean by a finite pointframework is a finite collection of points in a compact set E of a given Hausdorffdimension, where some, but not all the pairwise distances are specified. We encodethese frameworks in a rigorous way using combinatorial graphs. We then define asuitable notion of equivalence and embed the resulting equivalence classes in R m ,where m is the number of “essential” edges of the graph which encodes a givenframework. We then prove that if the Hausdorff dimension of the ambient set E islarger than a nontrivial threshold s k , then the m -dimensional Hausdorff content ofthe set of equivalences is positive. The precise formulation can be found in Theorem2.18 and Theorem 2.20 below.As the reader shall see below, a rigorous formulation of the Falconer type problemfor finite point frameworks naturally leads one to the notions of rigidity and otherinteresting concepts that combine combinatorial, topological and analytic concepts.The resulting symbiosis makes possible results that were not accessible using thepurely analytic methods that were employed in the cases of simplexes, chains andnecklaces. IGIDITY, GRAPHS AND HAUSDORFF DIMENSION 3 Definitions and statements of results
We shall encode finite point frameworks using combinatorial graphs. Let k ≥ K k +1 denote the complete graph with vertex set { , . . . , k + 1 } and edge setordered lexicographically. Let G k +1 ,m be a subgraph of K k +1 with k + 1 vertices and m edges inheriting the order. We define ij ∈ G k +1 ,m to mean that i < j and { i, j } is an edge of G k +1 ,m , and when ij ∈ G k +1 ,m ranges over all the edges of G k +1 ,m , itascends in the order of the edge set. Let | · | denote the Euclidean distance, k · k p the L p norm, dim( · ) denote the Hausdorff dimension and H s ( · ) denote the s -dimensionalHausdorff measure. Let L m ( · ) denote the Lebesgue measure of measurable subsetsof R m . Let A . B mean that for some constant C > A ≤ C · B , where C is independent of ǫ , δ and the summation index or integration variable (if used tobound the term). Moreover, A ≈ B means that A . B and B . A . Definition 2.1.
A ( k + 1) -tuple x in R d is a tuple x = ( x , x , . . . , x k +1 ) , x j ∈ R d . Definition 2.2. A framework of G k +1 ,m in R d is a pair ( G k +1 ,m , x ), where x is a( k + 1)-tuple in R d .A convenient way to specify distances is through the distance function which wenow define. Definition 2.3.
Given a graph G k +1 ,m we define the distance function f G k +1 ,m ( x )on x = ( x , . . . , x k +1 ) ∈ R d ( k +1) by f G k +1 ,m ( x ) = (cid:0) | x i − x j | (cid:1) ij ∈ G k +1 ,m . We also define the distance-squared function F G k +1 ,m ( x ) by F G k +1 ,m ( x ) = (cid:0) | x i − x j | (cid:1) ij ∈ G k +1 ,m . Definition 2.4 (Graph Distances) . The value f G k +1 ,m ( x ) is called the G k +1 ,m -distanceof x . When we restrict our domain to some set X ⊆ R d ( k +1) , we call f G k +1 ,m ( x ) a G k +1 ,m -distance on X and we say that x is a realization of this distance in X . Theset of G k +1 ,m -distances on X is f G k +1 ,m ( X ) and we denote it by ∆( G k +1 ,m , X ). Remark . Equivalence classes of frameworks (equivalent in the sense of the corre-sponding tuples having equal f G k +1 ,m -values) can be viewed as subsets of R m since m distances are specified in the sense of Definition 2.4. Given a graph, we ask whetherthere exists some 0 < s k < d such that any compact subset E of R d , d ≥
2, of Haus-dorff dimension larger than s k contains a positive m -dimensional measure (Hausdorffor Lebesgue, depending on the context) worth of equivalence classes of frameworksof the given graph, in other words, whether the set ∆( G k +1 ,m , E k +1 ) has positive IGIDITY, GRAPHS AND HAUSDORFF DIMENSION 4 (Hausdorff or Lebesgue) measure. Complete graphs in R d with at most d + 1 verticeswere comprehensively studied in [12]. In fact as we shall see later, when k ≤ d , theonly interesting case is the complete graph. Thus in this paper we consider graphswith k > d unless otherwise stated. Remark . The distance set ∆( G k +1 ,m , X ) depends on the numbering of the verticesand the order of the edges. Whereas the order of the edges is superficial, inducingonly a permutation in the components of the G k +1 ,m -distances, the numbering of thevertices can significantly change the G k +1 ,m -distance set. Consider X = { x } × R d ×· · · × R d and a graph G = G ′ ∪ G ′′ ∪ { e } where e is a bridge between G ′ and G ′′ .Then if we number the vertices of G ′ followed by those of G ′′ , we essentially capture G ′′ -distances only, whereas if we reverse the numbering order of the vertices of G we will capture G ′ -distances only. In the rest of this paper we take X = E k +1 forsome E ⊂ R d , so that the numbering of the vertices becomes superficial as well. Inparticular, the dimension of the G k +1 ,m -distance set and its Hausdorff (or Lebesgue)measure are independent of the vertex numbering and edge order.We define the notion of independence for subsets of the edge set of K k +1 and ofmaximal independence for subsets of the edge set of G k +1 ,m . We define the set ofgeneric tuples as the complement of the zero set of a certain polynomial. This notionis independent of the graph G k +1 ,m , depending only on the dimension d and thenumber of vertices k + 1.Let us use the following notation for our matrices: If a ij is a matrix, ( i, j ) ∈ I × J ,then for B ⊆ I, C ⊆ J , we defined a B,C to be the submatrix a ij with ( i, j ) ∈ B × C . Definition 2.7.
We say that x ∈ R d ( k +1) is a regular tuple of F G k +1 ,m if rank DF G k +1 ,m attains its global maximum at x . A framework ( G k +1 ,m , x ) is a regular framework if x is a regular tuple of F G k +1 ,m . Definition 2.8.
A subset H of the edge set of K k +1 is called independent in R d with respect to x ∈ R d ( k +1) if the row vectors of DF K k +1 ( x ) corresponding to H arelinearly independent. We call H independent in R d if there exists some x so that H is independent with respect to x , and x is said to be a witness to the independenceof H . We also call H a maximally independent (in R d ) subset of edges of G k +1 ,m when it is independent and it is not contained in a larger independent edge set of G k +1 ,m . Definition 2.9.
For any nonempty independent in R d set H of edges of K k +1 wedefine the polynomial P H ( x ) to be the sum of squares of | H | × | H | -minors of the IGIDITY, GRAPHS AND HAUSDORFF DIMENSION 5 submatrix of rows of DF K k +1 corresponding to edges of H . Thus, P H ( x ) = X A ⊂{ ,...,d ( k +1) }| A | = | H | (cid:12)(cid:12) det( DF K k +1 ( x ) H,A ) (cid:12)(cid:12) . Let X H denote the zero set of P H .We define the set of generic tuples of R d to be the complement of the zero set X of the polynomial P ( x ) defined by P ( x ) = Y H independent P H ( x ) . We call X the set of critical tuples of R d . Remark . We have X = ∪ H X H where the union is taken over all the edge sets H which are independent and the generic tuples are then equal to R d ( k +1) \ X . Moreover,if a set H of edges is independent then by Definition 2.9 it is generically independent,i.e. independent with respect to any generic x . In fact, the set of generic tuples isprecisely the set of tuples that simultaneously witness the independence of everyindependent edge set. Remark . The polynomial P ( x ) is nontrivial because every P H is nontrivialsince there is at least one witness x H for the independence H , which means that P H ( x H ) = 0. Thus X is a proper algebraic variety of dimensiondim X ≤ d ( k + 1) − . (2.1) Remark . It is immediate from the definitions that generic tuples are regulartuples. The other implication does not hold in general.
Definition 2.13.
A framework ( G k +1 ,m , x ) is called generic in R d if x is a generictuple in R d and it is called critical in R d if x is a critical tuple in R d .Our main results concern the dimension of the set ∆( G k +1 ,m , E k +1 ) and its Haus-dorff (or Lebesgue) measure. An important role is played by properties of the graph G k +1 ,m . In particular it is essential whether the graph is rigid or not.The key heuristic notion of this paper is that a graph G k +1 ,m is rigid in R d if oncethe m quantities t ij in | x i − x j | = t ij , ij ∈ G k +1 ,m are specified, the other distances | x i − x j | for ij G k +1 ,m can only take finitelymany values as the frameworks ( G k +1 ,m , x ) vary over the set of generic frameworks.For technical reasons, we use a more precise and flexible notion of rigidity describedbelow. A simple example that illustrates the technical obstacles one must contendwith is the following. Consider a quadrilateral in the plane with side-lengths 1 , , , IGIDITY, GRAPHS AND HAUSDORFF DIMENSION 6
This configuration is perfectly rigid in the heuristic sense, but it is not minimallyinfinitesimally rigid , as the reader will see, roughly because the rigidity in this caseis not stable under small perturbations.We now turn to precise definition.
Definition 2.14. An infinitesimal motion u = ( u , . . . , u k +1 ) in R d of G k +1 ,m at x is a ( k + 1)-tuple u of vectors u j ∈ R d such that DF G k +1 ,m ( x ) · u = 0 . The set of infinitesimal motions in R d of G k +1 ,m at x is the kernel of DF G k +1 ,m ( x ).Let us denote by V ( G k +1 ,m , x ) the set of infinitesimal motions in R d of G k +1 ,m at x .Let D ( x ) be the set of infinitesimal motions in R d of K k +1 at x . Remark . It is evident that D ( x ) ⊆ V ( G k +1 ,m , x ) since the system of equations DF G k +1 ,m ( x ) · u = 0 is included in DF K k +1 ( x ) · u = 0. Definition 2.16.
A framework ( G k +1 ,m , x ) is called infinitesimally rigid in R d when V ( G k +1 ,m , x ) = D ( x ).It is unnecessarily restrictive to require of a graph to have all its frameworks beinfinitesimally rigid. We shall only require it of generic frameworks. Definition 2.17.
A graph G k +1 ,m is called infinitesimally rigid in R d if all its genericframeworks are infinitesimally rigid. It is called minimally infinitesimally rigid in R d if it is infinitesimally rigid and no proper subgraph (on the same vertex set) isinfinitesimally rigid.2.1. Statements of results.Theorem 2.18.
Let G k +1 ,m be a connected graph that is minimally infinitesimallyrigid in R d , d ≥ and E is a compact subset of R d with dim E > d − k +1 . Then L m (∆( G k +1 ,m , E k +1 )) > . Remark . We shall in the proof of Theorem 2.18 that if G k +1 ,m is not connected,the proof naturally breaks into consideration of the connected components of thegraph. Theorem 2.20.
Let G k +1 ,m be a graph without isolated vertices and let E be acompact subset of R d , d ≥ with dim E > d − k +1 . Let H be a maximally independentsubset of edges of G k +1 ,m . Then dim ∆( G k +1 ,m , E k +1 ) = | H | IGIDITY, GRAPHS AND HAUSDORFF DIMENSION 7 and H | H | (cid:0) ∆( G k +1 ,m , E k +1 ) (cid:1) > . Remark . We stated Theorem 2.20 using the Hausdorff measure instead of theLebesgue measure because the edges not in H do not produce any further dimen-sionality (see Corollary 3.8 and Proposition 3.1). Remark . It should be pointed out that for our results we only work out thecase where k > d . Theorem 2.18 for k ≤ d was worked out in [12] and the betterthreshold dim E > dk +1 k +1 was obtained. Theorem 2.20 follows as a consequence, sincewhen k ≤ d , the only minimally infinitesimally rigid graph is the complete graph on k + 1 vertices and the independence condition of Theorem 2.20 is always satisfied(see Theorem 4.6) so H can be taken to be the edge set of G k +1 ,m and in thatcase Theorem 2.20 is a consequence of Theorem 2.18 by an application of Fubini’stheorem. Theorem 2.23 (Deforestation) . Let G k +1 ,m , m ≥ be a graph without isolatedvertices with a vertex v of degree and let G be the resulting graph when v isremoved from G k +1 ,m . Iterate this process obtaining a sequence G , . . . , G n , until G n has no more such vertices or when G n = K . Then in using Theorem 2.18 orTheorem 2.20 with G k +1 ,m , the dimensional threshold for E obtained may be takento be d − k + 1 − n . Remark . Thus trees disjoint from the rest of the graph except for the root canbe ignored by applying Theorem 2.23.We shall now see that our results are fairly sharp in the sense that the criticalexponent must in general tend to d as the number of vertices tends to infinity. Theorem 2.25.
Let E be a compact subset of R d , d ≥ of Hausdorff dimension s ∈ (0 , d ) . Then the conclusion of Theorem 2.18 and Theorem 2.20 does not ingeneral hold if s < d − ( d ) k . In dimension d = 2, the difference between the exponents in Theorem 2.18, The-orem 2.20 and Theorem 2.25 is not very large, 2 − k +1 versus 2 − k . In higherdimensions the gap increases, but Theorem 2.25 still shows that the correct criticalexponent must in general tend to d as the number of vertices tends to ∞ . IGIDITY, GRAPHS AND HAUSDORFF DIMENSION 8 Graph distances of subsets of R d Introduction.
Our goal is to prove that L m (∆( G k +1 ,m , E k +1 )) > s k with dim E > s k . Here we may assume G k +1 ,m isconnected, since we have Proposition 3.1. If G , . . . , G n are the connected components of G k +1 ,m on k , . . . , k n vertices respectively, then for cartesian products E k +1 where E ⊆ R d , we have ∆( G k +1 ,m , E k +1 ) = ∆( G , E k ) × · · · × ∆( G n , E k n ) . Proof.
It is clear that (after reordering the vertices if necessary) f G k +1 ,m = ( f G , . . . , f G n )where f G k +1 ,m , f G j are the corresponding distance functions of G k +1 ,m , G j . The resultfollows. (cid:3) Thus we only need to consider connected graphs, and requiring of G k +1 ,m to beconnected in Theorem 2.18 is not an essential restriction. If (3.1) does not hold, itmay be the case that the dimension of the G k +1 ,m -distance set is not full. Theorem2.20 then provides its Hausdorff dimension and positivity of the Hausdorff measure.We define the notion of congruency for tuples and frameworks. Definition 3.2.
Let x be a ( k + 1)-tuple in R d and define the set of tuples congruent to x to be M x = { ( T x , . . . , T x k +1 ) : T ∈ ISO( d ) } , (3.2)where ISO( d ) denotes the set of isometries of R d to itself. Definition 3.3.
We say that the framework ( G, x ) is congruent to the framework( G ′ , x ′ ) if G = G ′ and x is congruent to x ′ in the sense of Definition 3.2.We now describe some examples.3.2. Examples of ∆( G k +1 ,m , E k +1 ) . In the case where G k +1 ,m is the complete graph K on 2 vertices and E ⊆ R d , we recover the distance set of E ∆( K , E ) = {| x − y | : x, y ∈ E } . Now consider the complete graph K . Let d = 2 and E = R . We will directly showthat L (∆( K , R )) = 0. This is expected, as we will also show that dim ∆( K , R ) =5 and H (∆( K , R )) >
0. Split the tuples x ∈ R into two sets A and A , A j ⊂ R ,with x ∈ A iff ( x , x , x , x ) are in convex position and A the rest. We will workwith A but A is treated similarly. IGIDITY, GRAPHS AND HAUSDORFF DIMENSION 9 θ ψ
Figure 1.
A framework of K (dashed edge for emphasis).Let t ij denote the distance from the vertex i to j . By using Euler’s theorem forconvex quadrilaterals we may obtain the following equation, t = t + t − t + 2 t t cos( θ − ψ ) . (3.3)Here θ = cos − (cid:18) t + t − t t t (cid:19) , ψ = cos − (cid:18) t + t − t t t (cid:19) . Let t = g ( ˜ t ) where ˜ t = ( t , t , t , t , t ) . Thus, L (∆( K , A )) ≤ Z Z t = g ( ˜ t ) dt d ˜ t = 0 . This happens because equation (3.3) makes us integrate over the zero-dimensionalset (it is a singleton) t ∈ { g ( ˜ t ) } . Similarly we may obtain L (∆( K , A )) = 0.Thus K in d = 2 cannot possibly give us a dimensional threshold since even E = R has a K -distance set of zero measure. This happened because the graph had toomany edges. Not only ∆( K , R ) is a L -null set, but in fact its dimension is lessthan 6. By using Corollary 3.8, we find that its dimension is 5. Then using Theorem2.20 we see that it has positive H -measure.Now consider the following ‘double banana’ graph G , on R (dashed edge foremphasis, but it is in the edge set of G , ), IGIDITY, GRAPHS AND HAUSDORFF DIMENSION 10
Figure 2.
A framework of the ‘double banana’ G , graph.This is not an infinitesimally rigid graph. Each banana may be freely rotated aboutthe line joining the banana ends without altering the edge lengths. Yet it may notbe completed into a minimally infinitesimally rigid graph by adding edges becauseit contains redundant edges (the dashed one, for instance). The solid edges form amaximally independent set H of edges of G , , thus by an application of Theorem2.20 we obtain dim ∆( G , , E ) = 17 and H (∆( G , , E )) > E ⊆ R compactwith dim E > − .3.3. A sharp upper bound for the dimension of the distance set.
In thissection we determine the Hausdorff dimension of ∆( G k +1 ,m , R d ( k +1) ).If G k +1 ,m is a minimally infinitesimally rigid graph in R d , then from Corollary 4.10,it must have m = d ( k + 1) − (cid:18) d + 12 (cid:19) . (3.4)We may say that G k +1 ,m is minimally infinitesimally rigid in R d when its edge set isindependent and any edge added to G k +1 ,m turns the rows of DF G k +1 ,m into a linearlydependent set of vectors, as the next proposition shows. Proposition 3.4.
Let G k +1 ,m be a graph. Then the set of edges of G k +1 ,m is inde-pendent in R d and may not be enlarged while retaining independence if and only if G k +1 ,m is minimally infinitesimally rigid.Proof. If G k +1 ,m is minimally infinitesimally rigid in R d , then by definition for generictuples x ∈ R d ( k +1) , the kernel of DF G k +1 ,m ( x ) has the smallest dimension possible(in view of Theorem 4.7 and the dimension of the rotation group). Thus the edgeset can not be enlarged while retaining independence, since a larger independent setof edges would produce an even smaller kernel. IGIDITY, GRAPHS AND HAUSDORFF DIMENSION 11
On the other hand, assume G k +1 ,m has an independent in R d edge set that maynot be enlarged while retaining independence. We have just argued that G k +1 ,m cannot contain a minimally infinitesimally rigid proper subgraph. Assuming thenthat G k +1 ,m is not minimally infinitesimally rigid itself, for generic tuples x ∈ R d ( k +1) we know that V ( G k +1 ,m , x ) properly contains D ( x ). Thus there must be a set of edges H ⊂ K k +1 disjoint from those of G k +1 ,m with V ( H ∪ G k +1 ,m , x ) = D ( x ) . But that implies rank D F H ∪ Gk +1 ,m ( x ) > rank D F Gk +1 ,m ( x ), a contradiction to the as-sumption that G k +1 ,m has an edge set that may not be enlarged while retainingindependence. (cid:3) Proposition 3.5.
If the edge set of G k +1 ,m is independent in R d , then a minimallyinfinitesimally rigid (in R d ) graph G k +1 ,m containing G k +1 ,m exists.Proof. If k ≤ d , we just complete G k +1 ,m to the complete graph K k +1 since that isthe only rigid graph on k + 1 vertices in R d (see Theorem 4.6).Otherwise we pick x at random from a continuous distribution (say, the Gaussiandistribution) on R d ( k +1) . Since the set of critical frameworks is a proper algebraicvariety, it has Lebesgue measure zero and we have almost certainly (that is, withprobability 1) picked a generic framework.As long as the property of independence with respect to x is retained, we keepadding edges to G k +1 ,m until no more edges may be added. We then end up with agraph that is minimally infinitesimally rigid. (cid:3) Remark . The graph G k +1 ,m need not be unique. For instance, if G k +1 ,m is a tree,for large enough k there are many different minimally infinitesimally rigid graphs itmay complete to. Theorem 3.7.
Let G k +1 ,m be a connected graph and H a maximally independent in R d subset of edges of G k +1 ,m . Then dim ∆( G k +1 ,m , R d ( k +1) ) = | H | . Proof.
Let W ij denote the plane { x : x i = x j } . The map f G k +1 ,m is smooth away from W = S ij ∈ G k +1 ,m W ij and the rank of its total derivative does not exceed | H | there.Thus, since f G k +1 ,m has regular tuples away from W , we see that f G k +1 ,m ( R d ( k +1) \ W )has dimension | H | . Now consider f G k +1 ,m restricted to W ij . There the function issmooth away from W ij ∩ W kl for kl ∈ G k +1 ,m with kl = ij . The rank of the derivativeis less than or equal to | H | , so inductively f G k +1 ,m ( W ij ) has dimension less than orequal to | H | . (cid:3) IGIDITY, GRAPHS AND HAUSDORFF DIMENSION 12
Corollary 3.8. If G k +1 ,m is a connected graph that contains a minimally infinitesi-mally rigid (in R d ) subgraph G ∗ k +1 ,m ∗ , then dim ∆( G k +1 ,m , R d ( k +1) ) = m ∗ . Bounds on the number of noncongruent realizations.
Let G k +1 ,m beminimally infinitesimally rigid in R d and let x ∈ R d ( k +1) .We consider the set of preimages of f G k +1 ,m ( x ), N x = { y : f G k +1 ,m ( y ) = f G k +1 ,m ( x ) } . Define the equivalence relation y ∼ z by y ∈ M z , where M z is defined in (3.2) to bethe set of tuples congruent to z . The set N x is defined by the system of quadraticequations | y i − y j | = | x i − x j | , ij ∈ G k +1 ,m , and results on bounds of the Betti numbers of semi-algebraic varieties by Oleinikand Petrovskii, Thom and Milnor (see [29], [30], [31]) allow us to conclude that N x ,hence (since ISO( d ) has two connected components), N x / ∼ , has less than C d,k · dk connected components, for some C d,k >
0. For a better bound see [32]. In particular,when x is regular valued, we may conclude that there are at most C d,k · dk preimagesof f G k +1 ,m ( x ) up to congruences by Proposition 4.11. If x is not regular valued, itis possible for noncongruent preimages to lie in the same connected component, butin the argument to follow we will avoid critical frameworks. For our purposes weonly need the fact that if the critical tuples are removed, then the preimage set N x is finite up to congruences, with the bound independent of x .3.5. The proof of the dimensional threshold.
We prove Theorem 2.18.
Proof.
Let G = G k +1 ,m to ease subscript use. Fix E ⊂ R d compact, and let µ = µ s be a Borel probability measure supported on E , with Frostman exponent s . Thusthere exists some constant C µ > µ ( B ( x, r )) ≤ C µ r s for all balls B ( x, r ) ofradius r >
0, and we may choose s < dim E arbitrarily close to dim E . (See [23],Chapter 8 for the existence and properties of such measures). In particular, we maychoose s > d − k + 1 . (3.5)Let us first prove that the set of critical frameworks X is a null set for µ k +1 : µ k +1 ( X ) = 0 . Observe that µ k +1 is a Frostman measure of exponent s ( k + 1). This follows easilyfrom the fact that any ball B ( x , r ) is contained in a concentric cube Q = Q ×· · · × Q k +1 of side 2 r , where each Q j in turn is contained in a ball B ( x j , r ). Since µ k +1 ( Q ) = µ ( Q ) · · · µ ( Q k +1 ) . r s ( k +1) and s ( k +1) > dim( X ), which we may assume IGIDITY, GRAPHS AND HAUSDORFF DIMENSION 13 since the dimension of E is big enough (by (2.1) and (3.5)) and using the fact thatsets of positive measure of a Frostman measure have dimension greater or equal tothe Frostman exponent (see Lemma 4.15), we conclude that X is a null set for µ k +1 .Let δ > µ k +1 ( E k +1 \ X δ ) > / X δ the δ -neighborhood of X defined by X δ = { y ∈ R d ( k +1) : | y − x | < δ } . Sucha δ exists since X is closed and X = T δ> X δ . We want to avoid getting close to X because our named constants in the arguments to follow blow up near it. Let A = E k +1 \ X δ and let ν ( t ) be the pushforward of µ k +1 ( x ) by f G | A , that is, for anymeasurable function g ( t ) the integral R gdν is defined by Z g ( t ) dν ( t ) = Z A g (cid:0) ( | x i − x j | ) ij ∈ G (cid:1) dµ ( x ) · · · dµ ( x k +1 ) . From now on we shall write dµ k +1 ( x ) for dµ ( x ) · · · dµ ( x k +1 ). We shall show that ν ( R m ) > ν ∈ L ( R m ) implying L m (supp ν ) > ν ⊂ ∆( G, E k +1 ). For the first claim, ν ( R m ) = µ k +1 ( E k +1 \ X δ ) > / . Now we will prove that ν ∈ L ( R m ). Let ν ǫ = φ ǫ ∗ ν , where φ ǫ ( t ) = ǫ − m φ ( ǫ − t ) and φ ∈ C ∞ c ( R m ) is a nonnegative radial function with R φ = 1, φ ≤ φ ⊂ B (0 , χ the characteristic function of a set. We have, ν ǫ ( t ) = Z A ǫ − m φ (cid:0) ǫ − ( f G ( x ) − t ) (cid:1) dµ k +1 ( x ) ≤ Z A ǫ − m χ (cid:8)(cid:12)(cid:12) f G ( x ) − t (cid:12)(cid:12) < ǫ (cid:9) dµ k +1 ( x ) . (3.6)By Lemma 4.13, as ǫ → ν ǫ → ν in the weak topology of the dual of C ( R m ). We conclude that lim inf k ν ǫ k ≥ k ν k from Lemma 4.14, and thus itsuffices to bound lim inf k ν ǫ k . By an application of the triangle inequality on (3.6)we have, k ν ǫ k ≤ ǫ − m Z Z A Z A χ (cid:8)(cid:12)(cid:12) f G ( x ) − t (cid:12)(cid:12) < ǫ (cid:9) χ (cid:8)(cid:12)(cid:12) f G ( y ) − t (cid:12)(cid:12) < ǫ (cid:9) dµ k +1 ( x ) dµ k +1 ( y ) d t ≤ ǫ − m Z | t | < ǫ d t · Z A Z A χ (cid:8)(cid:12)(cid:12) f G ( x ) − f G ( y ) (cid:12)(cid:12) < ǫ (cid:9) dµ k +1 ( x ) dµ k +1 ( y ) . ǫ − m Z A Z A χ (cid:8)(cid:12)(cid:12) f G ( x ) − f G ( y ) (cid:12)(cid:12) < ǫ (cid:9) dµ k +1 ( x ) dµ k +1 ( y ) . Consider y to be fixed now. Note that it is a regular tuple since it belongs to A . Thecondition that the images of x and y are ǫ -close is giving us . dk open sets of R d ( k +1) where x may lie (as explained in Section 3.4). Let U , . . . , U l be those open sets, and IGIDITY, GRAPHS AND HAUSDORFF DIMENSION 14 let Z = { z , . . . , z l } be such that each z j ∈ U j ∩ A (potentially picking less than l tuples, if some intersections are empty). Denote by O d ( R ) the group of rotations of R d . Cover the compact Riemannian manifold O d ( R ) by ǫ -balls T ǫ , . . . , T ǫK ( ǫ ) of finite(uniformly in ǫ ) overlap with centers g , . . . , g K ( ǫ ) .Then the set { x ∈ A : | f G ( x ) − f G ( y ) | < ǫ } is a subset of the set [ z ∈ Z K ( ǫ ) [ k =1 { x ∈ A : | ( x i − x j ) − g k ( z i − z j ) | < cǫ, ∀ ij ∈ G } for some c > y (as Proposition 4.11 shows, f G isbiLipschitz in U , . . . , U l , once congruences are identified). Since A is a compact set, c attains a maximum value, so pick such c to lift the dependence on y .Since | Z | . dk , it follows that, k ν ǫ k . ǫ − m Z A X z ∈ Z K ( ǫ ) X k =1 Z A χ {| ( x i − x j ) − g k ( z i − z j ) | < cǫ, ∀ ij ∈ G } dµ k +1 ( x ) dµ k +1 ( y ) . dk ǫ − m K ( ǫ ) X k =1 Z A Z A χ {| ( x i − x j ) − g k ( y i − y j ) | < cǫ, ∀ ij ∈ G } dµ k +1 ( x ) dµ k +1 ( y ) . The volume of the ǫ -balls of O d ( R ) is ≈ ǫ dim O d ( R ) = ǫ d ( d − / . In what follows,we estimate the value of a function at a point by twice the average of that functionaround an ǫ -ball and use the fact that they cover O d ( R ) with finite overlap to obtain, k ν ǫ k . ǫ − m K ( ǫ ) X k =1 ǫ d ( d − / Z T ǫk Z A Z A χ {| ( x i − x j ) − g ( y i − y j ) | < cǫ, ∀ ij ∈ G } dµ k +1 ( x ) dµ k +1 ( y ) dg . ǫ − dk Z O d ( R ) Z A Z A χ {| ( x i − x j ) − g ( y i − y j ) | < cǫ, ∀ ij ∈ G } dµ k +1 ( x ) dµ k +1 ( y ) dg . (3.7)Here we used (3.4) to get m + d ( d − / dk . For g ∈ O d ( R ), define ν g by Z f ( x ) dν g ( x ) = Z Z f ( u − gv ) dµ ( u ) dµ ( v ) . Let G ′ ⊂ G be a spanning tree. We continue (3.7) with k ν ǫ k . ǫ − dk Z O d ( R ) Z A Z A χ {| ( x i − x j ) − g ( y i − y j ) | < cǫ, ∀ ij ∈ G ′ } dµ k +1 ( x ) dµ k +1 ( y ) dg . (3.8) IGIDITY, GRAPHS AND HAUSDORFF DIMENSION 15
Using Lemma 3.12 (to be proved below) on (3.8) we obtain k ν ǫ k . Z Z ν k +1 g ( x ) dxdg . (3.9)Theorem 3.10 shows this integral to be finite for s > d − k +1 , concluding the proofthat ν ∈ L ( R m ) and thus showing that L m (∆( G, E k +1 )) > (cid:3) We may now go a step further and prove Theorem 2.20.
Proof. If G k +1 ,m is any connected graph, and H is a maximally independent subsetof the edge set of G k +1 ,m , we may complete H to a minimally infinitesimally rigidgraph H k +1 ,m (see Proposition 3.5). By using Theorem 2.18, we obtain the nontriv-ial exponent d − k +1 for ∆( H k +1 ,m , E k +1 ) to have positive Lebesgue measure. Weproject ∆( H k +1 ,m , E k +1 ) → ∆( H, E k +1 ) by ( t ij ) ij ∈ H k +1 ,m ( t ij ) ij ∈ H to show that∆( H, E k +1 ) has positive Lebesgue measure by Fubini.Now, projecting ∆( G k +1 ,m , E k +1 ) → ∆( H, E k +1 ) by ( t ij ) ij ∈ G k +1 ,m ( t ij ) ij ∈ H shows that ∆( G k +1 ,m , E k +1 ) has positive H | H | -measure, because the projection isLipschitz. Lastly, Theorem 3.7 shows that dim ∆( G k +1 ,m , E k +1 ) = | H | .Moreover, if G k +1 ,m has connected components G , . . . G n , we will obtain a maxi-mally independent subset H = H ∪ · · · ∪ H n of G k +1 ,m where each H j is a maximallyindependent subset of G j for j = 1 , . . . , n . Using Proposition 3.1 and what we justargued for connected graphs, we again obtain a positive | H | -Hausdorff measuresworth of distances. (cid:3) We now prove Theorem 2.23.
Proof.
As before let G = G k +1 ,m to avoid notational clutter. Let σ t denote thesurface measure of the sphere tS d − ⊂ R d of radius t > φ ǫ ( x ) = ǫ − d φ ( ǫ − x ) where φ ∈ C ∞ c ( R d ) is a nonnegative radial function with φ = 1 on B (0 , φ ≤ φ ⊂ B (0 , σ ǫt = φ ǫ ∗ σ t . We note that c · χ { y : || y |− t | <ǫ } ( x ) ≤ ǫσ ǫt ( x ) ≤ C · χ { y : || y |− t | < ǫ } ( x ) (3.10)for some constants c > C > φ only. Without loss of generalityassume v = k + 1 and that { k, k + 1 } is an edge of G . Let t = ( t ij ) ij ∈ G , t ij ∈ (0 , + ∞ ),and consider the function (on the domain of t just mentioned),Λ ǫG,µ ( t ) = Z Y ij ∈ G σ ǫt ij ( x i − x j ) dµ ( x ) · · · dµ ( x k +1 ) . Using (3.10) we see that obtaining a bound k Λ ǫG,µ k ≤ M with M independent of ǫ is equivalent to obtaining a bound for k ν ǫ k independent of ǫ , in particular showing IGIDITY, GRAPHS AND HAUSDORFF DIMENSION 16 that ν ∈ L ( R m ). Write thenΛ ǫµ ( t ) = Z Y ij ∈ G,ij = k ( k +1) σ ǫt ij ( x i − x j ) dµ ( x ) · · · dµ ( x k − ) · Z σ ǫt k ( k +1) ( x k − x k +1 ) dµ ( x k ) dµ ( x k +1 )= Z Y ij ∈ G,ij = k ( k +1) σ ǫt ij ( x i − x j ) dµ ( x ) · · · dµ ( x k − ) ( σ ǫt k ( k +1) ∗ µ )( x k ) dµ ( x k )From Lemma 4.16 we know that k σ ǫt ∗ µ k L ( µ ) . E ′ ⊂ E with µ ( E ′ ) > k σ ǫt ∗ µ k L ∞ ( E ′ ,µ ) . µ ′ the restriction of µ to E ′ , we obtain a Frostman measure of the sameexponent. Denote by G ′ the graph G with the vertex v removed. Thus we have nowΛ ǫG,µ ′ ( t ) . Z Y ij ∈ G ′ σ ǫt ij ( x i − x j ) dµ ′ ( x ) · · · dµ ′ ( x k +1 )= Λ ǫG ′ ,µ ′ ( t )The rest of the proof proceeds as in Theorem 2.18, with µ ′ in place of µ and G ′ inplace of G . (cid:3) The natural measure ν g on E − gE . We denote by S d − the ( d − R d . We will need the following result. Theorem 3.9 (Wolff-Erdo˘gan Theorem) . Let µ be a compactly supported Borel mea-sure in R d . Then, for s ≥ d/ and ǫ > , Z S d − | ˆ µ ( tω ) | dω ≤ C ǫ I s ( µ ) t ǫ − γ ( s,d ) , with γ ( s, d ) = ( d +2 s − / if d/ ≤ s ≤ ( d +2) / and γ ( s, d ) = s − for s ≥ ( d +2) / where I s ( µ ) is the s -energy of µ , I s ( µ ) = RR | x − y | − s dµ ( x ) dµ ( y ) . For s ≤ ( d + 2) /
2, see Wolff [22] for d = 2, and Erdo˘gan [8] for d ≥
3. Forthe case s ≥ ( d + 2) /
2, see Sj¨olin [24]). See [23] Chapter 8 for the definition andrelevant properties of s -energy, we are only interested in the fact that it will be afinite number. Theorem 3.10 (Natural measure on E − gE ) . Let k ≥ and let E ⊂ R d , d ≥ be acompact set with dim E > d . Let µ be a Borel probability measure on E of Frostman IGIDITY, GRAPHS AND HAUSDORFF DIMENSION 17 exponent s < dim E with s satisfying ( s > d (4 k − k +2 for d < dim E ≤ d +22 s > kd − k +1 for d +22 < dim E .
Let g be a rotation, g ∈ O d ( R ) and define the measure ν g by Z f dν g = Z Z f ( u − gv ) dµ ( u ) dµ ( v ) . (3.11) Then the integral
Z Z ν k +1 g ( x ) dxdg is a finite quantity and in particular ν g is absolutely continuous for g -a.e.Remark . The threshold for s when d < dim E ≤ dim d + 1 is not useful unless d = 2 or d = 3 , k = 1 , k = 1, since in that case d (4 k −
1) + 24 k + 2 ≥ d + 22 ≥ dim E .
In particular, below is a table for the readers convenience.Exponents for s when d < dim E ≤ d + 22 d = 2 s > k k + 1 d = 3 , k = 1 , s > k − k + 2 d > , k = 1 s > d d, k are not listed on this table then we havedim E > s > kd − k + 1 . Proof.
Let ψ ≥ { ξ ∈ R d : 1 / ≤ | ξ | ≤ } ,identically equal to 1 in { ξ ∈ R d : 1 ≤ | ξ | ≤ } with √ ψ also smooth. Let ψ j ( ξ ) = ψ (2 − j ξ ). Moreover, we require P + ∞ j = −∞ ψ j ( ξ ) = c , for a suitable constant c >
0, for all ξ = 0 (see [23] § ψ ). Let ν g,j denote the j -th Littlewood-Paleypiece of ν g defined by ˆ ν g,j ( ξ ) = ˆ ν g ( ξ ) ψ j ( ξ ). Since ν g is a finite measure, in boundingthe pieces, we may assume that j is bounded from below. Using the Littlewood-Paley IGIDITY, GRAPHS AND HAUSDORFF DIMENSION 18 decomposition of ν g , we may write R ν k +1 g ( x ) dx as Z X j ,...,j k +1 ν g,j ( x ) · · · ν g,j k +1 ( x ) dx . This is bounded above by( k + 1)! Z X j ≥ j ≥···≥ j k +1 ν g,j ( x ) · · · ν g,j k +1 ( x ) dx . Now using Plancherel, we see that since ˆ ν g,j ∗ · · · ∗ ˆ ν g,j k +1 is supported on scale2 j + · · · + 2 j k +1 ≤ j +1 while ˆ ν g,j is supported on an annulus of scale 2 j , the sumvanishes if j − j > j large. Thus it suffices to consider the case j = j (thecase j = j + 1 is similarly treated) and to look at the sum X j = j ≥ j ≥···≥ j k +1 Z ν g,j ( x ) ν g,j ( x ) · · · ν g,j k +1 ( x ) dx . (3.12)From the definition of ν g,j it follows that ν g,j = µ j ( −· ) ∗ µ j ( g · ), where ˆ µ j = ˆ µ p ψ j .By Young’s inequality, k ν g,j k ∞ ≤ k µ j k · k µ j k ∞ . Trivially k µ j k ≤ µ is a probability measure. Also | µ j ( x ) | = 2 dj | µ ∗ dp ψ ( − j x ) | ≤ C N dj Z (1 + 2 j | x − y | ) − N dµ ( y ) ≤ C ′ N j ( d − s ) for any N > µ is a Frostman measure on E . Using this estimate on the termscorresponding to the indices j , . . . , j k +1 we can bound (3.12) by a constant multipleof Z X j X j ≥ j ≥···≥ j k +1 ( j + ··· + j k +1 )( d − s ) ν g,j ( x ) dx . Z X j j ( k − d − s ) ν g,j ( x ) dx . It follows that
Z Z ν k +1 g ( x ) dxdg . X j j ( k − d − s ) · Z Z ν g,j ( x ) dxdg . We will show that
R R ν g,j ( x ) dxdg . j ( d − s ) − jγ ( s,d ) where the quantity γ ( s, d ) isdefined in Theorem 3.9, which completes the proof. IGIDITY, GRAPHS AND HAUSDORFF DIMENSION 19
Since we have ˆ ν g,j = ˆ µ j ( ξ )ˆ µ j ( gξ ), via Plancherel we obtain Z Z ν g,j ( x ) dxdg = Z Z | ˆ µ j ( ξ ) | | ˆ µ j ( gξ ) | dξdg = Z ∞ Z S d − | ˆ µ j ( tω ) | (cid:18)Z | ˆ µ j ( gtω ) | dg (cid:19) t d − dωdt . Since O d ( R ) acts transitively on the sphere, the quantity in the parentheses is con-stant in ω , and in particular it is a constant multiple of R | ˆ µ j ( tω ′ ) | dω ′ . Thus wehave that Z Z ν g,j ( x ) dxdg = C Z (cid:18)Z S d − | ˆ µ j ( tω ) | dω (cid:19) t d − dt = C Z (cid:18)Z S d − | ˆ µ ( tω ) | ψ (2 − j tω ) dω (cid:19) t d − dt = C ′ Z j +2 j − (cid:18)Z S d − | ˆ µ ( tω ) | dω (cid:19) t d − dt . Since we are summing over j and the intervals [2 j − , j +2 ] have finite overlap witheach other, we may as well bound P R R ν g,j ( x ) dxdg by a constant multiple of Z j +1 j (cid:18)Z S d − | ˆ µ ( tω ) | dω (cid:19) t d − dt . The proof is finished by using Theorem 3.9, showing that
R R ν g ( x ) dxdg < ∞ . (cid:3) The proof is now complete up the proof of Lemma 3.12 and the geometric resultsin Section 4. We prove the lemma below. The geometric results are established inSection 4.
Lemma 3.12.
Let G ′ k +1 ,m be a tree. Then for small enough ǫǫ − dk Z O d ( R ) Z A Z A χ {| ( x i − x j ) − g ( y i − y j ) | < cǫ : ij ∈ G ′ k +1 ,m } dµ k +1 ( x ) dµ k +1 ( y ) dg (3.13) is bounded by a constant multiple of Z Z ν k +1 g ( x ) dxdg . IGIDITY, GRAPHS AND HAUSDORFF DIMENSION 20
Proof.
First we bound (3.13) by ǫ − dk Z Z Z χ {| ( x − x j ) − g ( y − y j ) | < ( k + 1) cǫ : j > } dµ k +1 ( x ) dµ k +1 ( y ) dg . (3.14)This is accomplished as follows: Fix ij ∈ G ′ k +1 ,m and let ( x , x l , . . . , x l p , x i , x j ) bea path of length p + 1, from x to x j in G ′ k +1 ,m . Set l = 1 and l p +1 = i , l p +2 = j .Using the triangle inequality we see that the set { ( x , y ) : | ( x i − x j ) − g ( y i − y j ) | < cǫ } is contained in the set { ( x , y ) : p +1 X f =1 | ( x l f − x l f +1 ) − g ( y l f − y l f +1 ) | < ( p + 1) cǫ } which is contained in { ( x , y ) : | ( x − x j ) − g ( y − y j ) | < ( p + 1) cǫ } . Since k ≥ p we conclude that (3.13) is bounded by (3.14). Using now the naturalmeasure ν g we write (3.14) as ǫ − dk Z Z · · · Z χ {| z − z j | < cǫ : j > } dν g ( z ) · · · dν g ( z k +1 ) dg . Now it is obvious that taking ǫ → ν g and thedominated convergence theorem, we may bound (3.14) by a constant multiple of Z Z ν k +1 g ( z ) dzdg finishing the proof. (cid:3) Geometric results
For d ≥ ≤ q ≤ d we show that the edge set of K q +1 is independent in R d . We show that infinitesimal rigidity of a fixed graph G k +1 ,m is a generic property,either holding for all generic frameworks, or none of them. We count the numberof edges a minimally infinitesimally rigid graph must have. The behavior of thedistance function near regular tuples is investigated. Our approach follows closelythat of [26]. See also [28] for motivation and examples. IGIDITY, GRAPHS AND HAUSDORFF DIMENSION 21
Generic Frameworks.
In this section we prove various results for genericframeworks in R d . Some of the statements are for regular frameworks, but as notedin Remark 2.12, generic frameworks are regular.The following lemma, while technically obvious, serves to remind the reader of theform that DF G k +1 ,m takes, which will be useful in subsequent proofs in this section. Lemma 4.1.
Fix d ≥ . We have DF G k +1 ,m ( x ) · u = 0 if and only if u is a solutionto the following system of m equations in d ( k + 1) variables: ( x i − x j ) · ( u i − u j ) = 0 , for ij ∈ G k +1 ,m . (4.1) Proof.
Since F G k +1 ,m is a function R d ( k +1) → R m , DF G k +1 ,m is a m × d ( k + 1) matrixwith rows corresponding to edges ij ∈ G k +1 ,m and columns corresponding to thescalar components of x . The ij -th row is equal to the following vector(0 , . . . , , d ( i − di z }| { x i − x j ) , , . . . , , d ( j − dj z }| { − x i − x j ) , , . . . , . Here every component x i − x j is also a vector ( x , . . . , x k +1 ∈ R d are vectors). Thuswe can see that DF G k +1 ,m ( x ) · u = 0 is equivalent to (4.1). (cid:3) Proposition 4.2. If H ⊆ K k +1 is an independent (in R d ) set of edges then x ∈ X H ifand only if the rows of DF K k +1 ( x ) corresponding to edges of H are linearly dependent.Moreover if H ⊂ H ′ ( H, H ′ independent), then X H ⊆ X H ′ .Proof. Since the rank of the matrix is less than | H | , the | H | row vectors are linearlydependent. Conversely, if the rows are linearly dependent every minor has to be zerosince all the submatrices will satisfy the same dependence.Moreover, if H ⊂ H ′ then the matrix corresponding to H ′ will have rank at leastthat of the one for H . (cid:3) Theorem 4.3.
The set of generic tuples in R d is an open dense set of full Lebesguemeasure. Moreover every independent (in R d ) set H is in fact independent in R d with respect to any generic tuple in R d .Proof. Note that to each polynomial P H corresponds at least one tuple x for which P H is nonzero, thus the zero sets X H are proper algebraic varieties. Thus the set ofgeneric tuples of G is nonempty, and in particular open dense of full measure (sincethe complement X is of codimension at least 1, as a proper algebraic variety).The independence of H for any generic tuple follows from the definition of gener-icity. In particular if x is generic then x X , thus x X H . By Proposition 4.2 itfollows that H is independent with respect to x . (cid:3) Lemma 4.4. If A is an invertible affine transformation of R d and we set A x =( Ax , . . . , Ax k +1 ) , then we have that AX = X . Thus invertible affine transformationspreserve genericity. IGIDITY, GRAPHS AND HAUSDORFF DIMENSION 22
Proof. If Ax = Bx + b where B is an invertible linear transformation and b a vector,then since the row vectors of DF K k +1 ( x ) contain the entries ± ( x i − x j ), we see thatthe row vectors of DF K k +1 ( A x ) contain the entries ± ( Ax i − Ax j ) = ± B ( x i − x j ).In particular we see that the rows of DF K k +1 ( x ) corresponding to H are linearlyindependent if and only if those of DF K k +1 ( A x ) are. Using Proposition 4.2 weconclude that affine transformations preserve genericity. (cid:3) Definition 4.5.
The tuple x = ( x , . . . , x k +1 ) is said to be in general position in R d if for every set J ⊆ { , . . . , k + 1 } with | J | ≤ d + 1 we have that { x j : j ∈ J } isaffinely independent. Theorem 4.6.
Assume q ≤ d . The edge set of K q +1 is then independent in R d , infact with respect to any tuple in general position.Proof. Let x be in general position and assume that the rows of DF K q +1 ( x ) arelinearly dependent, say X ≤ i
Let x be a generic tuple in R d . Then x is in general position in R d .Proof. Assume x is not in general position. Thus for some 1 ≤ q ≤ d without lossof generality we may assume { x , . . . , x q +1 } are affinely dependent with { x , . . . , x q } affinely independent. As proven in Theorem 4.6, the edge set of K q +1 is independent.Let A be the affine transformation taking each x j , 1 ≤ j ≤ q to e j , the standard j -th basis vector (using Lemma 4.4). Then by affine dependence we must have Ax q +1 = t e + · · · + t q e q with t + · · · + t q = 0. Setting s ij = − t i t j for 1 ≤ i < j ≤ q and s i ( q +1) = t i for 1 ≤ i ≤ q we easily check that for any 1 ≤ a ≤ q + 1, X ia ∈ K q +1 s ia ( x a − x i ) + X aj ∈ K q +1 s aj ( x a − x j ) = 0 . Thus the edge set of K q +1 is not independent with respect to ( x , . . . , x q +1 ) andso by Theorem 4.3 we conclude that ( x , . . . , x q +1 ) is not a generic tuple. We nownote that if x were a generic tuple then ( x , . . . , x q +1 ) would also be a generic tuple,simply because we are removing d ( k + 1) − d ( q + 1) column vectors from DF K k +1 IGIDITY, GRAPHS AND HAUSDORFF DIMENSION 23 to test for the genericity of ( x , . . . , x q +1 ). This contradicts what we have foundtherefore x is not generic. (cid:3) Lemma 4.8. If x is in general position in R d then dim D ( x ) = (cid:0) d +12 (cid:1) .Proof. First let us show that if γ ( t ) : ( − , → M x is a smooth curve with γ (0) = x ,then γ ′ (0) ∈ D ( x ). This follows from the fact that the composition F ( γ ( t )) isconstant, so by the chain rule DF ( γ (0)) · γ ′ (0) = 0. Note that dim M x = (cid:0) d +12 (cid:1) ,giving us dim D ( x ) ≥ (cid:18) d + 12 (cid:19) . For the reverse inequality, we will show that any infinitesimal motion u ∈ D ( x )projects injectively to an infinitesimal motion ˜ u of V ( K d +1 , ˜ x ) where ˜ u = ( u , . . . , u d +1 )and ˜ x = ( x , . . . , x d +1 ). By the rank-nullity theorem and Theorem 4.6,dim V ( K d +1 , ˜ x ) = d ( d + 1) − (cid:18) d + 12 (cid:19) = (cid:18) d + 12 (cid:19) , establishing the reverse inequality and completing the proof.Thus it remains to show injectivity. Let u , v ∈ D ( x ) and assume ˜ u = ˜ v , that is, u i = v i for 1 ≤ i ≤ d + 1. Since the space D ( x ) is a vector space, for w = u − v wehave ( w i − w j ) · ( x i − x j ) = 0 , for ij ∈ K k +1 . Now if i = 1 , . . . , d we have w i = 0, so that for any j > d we have w j · ( x i − x j ) = 0 , for i = 1 , . . . , d . But that means w j is perpendicular to d linearly independent vectors, since x is ingeneral position, thus w j = 0 as well, and u = v . (cid:3) Theorem 4.9.
Let ( G k +1 ,m , x ) be an infinitesimally rigid framework in R d with x generic. Then for all generic tuples x , the frameworks ( G k +1 ,m , x ) are infinitesimallyrigid in R d .Proof. Since V ( G k +1 ,m , x ) = D ( x ), combine Lemma 4.8 and Theorem 4.7 to obtaindim V ( G k +1 ,m , x ) = (cid:18) d + 12 (cid:19) . Since V ( G k +1 ,m , x ) = ker DF G k +1 ,m ( x ) by the rank-nullity theorem we obtaindim V ( G k +1 ,m , x ) = d ( k + 1) − rank DF G k +1 ,m ( x ) . (4.2)Combining these two equations we find thatrank DF G k +1 ,m ( x ) = d ( k + 1) − (cid:18) d + 12 (cid:19) . IGIDITY, GRAPHS AND HAUSDORFF DIMENSION 24
Since generic tuples have the same rank (by Theorem 4.3), by using Equation(4.2) with x in place of x we see that dim V ( G k +1 ,m , x ) = (cid:0) d +12 (cid:1) , implying that V ( G k +1 ,m , x ) = D ( x ), so that ( G k +1 ,m , x ) is infinitesimally rigid. (cid:3) Corollary 4.10.
A minimally infinitesimally rigid graph G k +1 ,m in R d satisfies m = d ( k + 1) − (cid:18) d + 12 (cid:19) . Proof.
Let G k +1 ,m be minimally infinitesimally rigid. Then m ≥ rank DF G k +1 ,m ( x ) = d ( k + 1) − (cid:0) d +12 (cid:1) , as the proof of Theorem 4.9 shows. Let ( G k +1 ,m , x ) be a regularframework. If we assume m > d ( k + 1) − (cid:0) d +12 (cid:1) , there must be a subset H of edgesof G k +1 ,m such that rank DF H = d ( k + 1) − (cid:18) d + 12 (cid:19) = | H | about x , therefore H is infinitesimally rigid about x . Since that is an open set, byTheorem 4.9 the generic behavior of H is determined, thus H is infinitesimally rigid,which is a contradiction since H has less edges than G k +1 ,m . (cid:3) Proposition 4.11.
Let G k +1 ,m be a minimally infinitesimally rigid graph in R d and ( G k +1 ,m , x ) be a regular framework. Then there exists some open neighborhood U of x and an embedded m -dimensional submanifold M ⊂ U that contains x , with F G k +1 ,m restricted on M a diffeomorphism onto its image. Moreover if x ∈ U , letting N x = { y : F G ( y ) = F G ( x ) } denote the level curves, we have N x ∩ U = { ( T x , . . . , T x k +1 ) : T ∈ ISO( d ) } ∩ U .
The same also hold for f G k +1 ,m . x MN x ∩ UU Figure 3.
The local behavior of the distance function at a regular tuple.
IGIDITY, GRAPHS AND HAUSDORFF DIMENSION 25
Proof.
Since rank DF G k +1 ,m ( x ) = m is maximal, it stays maximal around an openneighborhood U of x . The Inverse Function Theorem yields local coordinates ( p , q )at x such that F G k +1 ,m ( p , q ) = p . The manifold M is the image of ( p , m . Using Corollary 4.10 and the fact that dim ISO( d ) = (cid:0) d +12 (cid:1) , theother claims follow. (cid:3) Remark . We justly say that the regular frameworks of a minimally infinitesi-mally rigid graph are locally uniquely realizable, in the sense that modulo isometriesthe distance function is a local diffeomorphism as Proposition 4.11 shows.4.2.
Useful Lemmas.
Here we collect the rest of lemmas that were used, that arenot related to graph rigidity.
Lemma 4.13.
Let φ ∈ C ∞ c ( R m ) be a nonnegative radial function with R φ = 1 , φ ≤ and supp φ ⊂ { t : | t | ≤ } and for ǫ > set φ ǫ ( t ) = ǫ − m φ ( ǫ − t ) . Let ν ( t ) be aBorel probability measure. Then ν ǫ = φ ǫ ∗ ν converges weakly-* to ν .Proof. Let f ∈ C ( R m ) be a nonnegative function that vanishes at infinity. Then Z f dν ǫ = Z ( f ∗ φ ǫ ) dν . It is a well known result of mollifiers that f ∗ φ ǫ → f pointwise and by the DominatedConvergence Theorem we may conclude that R f dν ǫ → R f dν . (cid:3) Lemma 4.14.
Let V be a normed vector space and V ∗ its dual equiped with theoperator norm. If ν n → ν weakly-* in V ∗ then lim inf k ν n k ≥ k ν k .Proof. lim n →∞ inf k ≥ n sup k x k =1 h x, ν k i ≥ lim n →∞ sup k x k =1 inf k ≥ n h x, ν k i ≥ sup k x k =1 h x, ν i . (cid:3) Lemma 4.15. If µ is a measure on R n and C µ > a constant with µ ( B ( x, r )) ≤ C µ r s for some s > and all r > and x ∈ R n , then for any measurable subset A of R n with µ ( A ) > we have dim A ≥ s .Proof. Let U j be open balls of radius r j covering A . We then have0 < µ ( A ) ≤ ∞ X j =1 µ ( U j ) ≤ C µ ∞ X j =1 r sj . Taking the infimum over all such collections U j we obtain that H s ( A ) > (cid:3) IGIDITY, GRAPHS AND HAUSDORFF DIMENSION 26
Lemma 4.16.
Let σ t denote the surface measure of the sphere tS d − ⊂ R d of radius t centered at . Let φ ǫ ( x ) = ǫ − d φ ( ǫ − x ) and φ ∈ C ∞ c ( R d ) is a nonnegative radialfunction with R φ = 1 , φ ≤ and supp φ ⊂ B (0 , . Let σ ǫt = φ ǫ ∗ σ t . Let µ be aFrostman measure on E ⊂ R d , E compact, with Frostman exponent s > d +12 . Thenthere exists a constant C t > independent of ǫ with k σ ǫt ∗ µ k L ( µ ) < C t . Proof.
We use Plancherel and the stationary phase of the sphere, (see [23]), that tellsus that for ξ of large norm and some c > | b σ ǫt ( ξ ) | ≤ ct d − | ξ | − d − , to obtain that for some C > E , Z σ ǫt ∗ µ ( x ) dµ ( x ) = Z b σ t ( ξ ) b φ ( ǫξ ) | b µ | ( ξ ) dξ . Z | ξ | − d − | b µ | ( ξ ) dξ . Z Z | x − y | − d +12 dµ ( x ) dµ ( y ) . Z Z + ∞ C µ n x : | x − y | < λ − d +1 o dλdµ ( y ) . Z + ∞ C λ − d +1 s dλ The last integral is finite by assumption. (cid:3) Proof of Theorem 2.25
Let q be a positive integer and define E q to be the q − ds -neighborhood of q n Z d ∩ [0 , q ] d o with s ∈ (cid:0) d , d (cid:1) to be determined later. It is known (see e.g. [9]) that if we choose q = 2, q i +1 > q ii , then the Hausdorff dimension of E = ∩ i E q i is equal to s . Lemma 5.1.
The number of congruence classes of frameworks with k + 1 verticesin Z d ∩ [0 , q ] d is bounded above by Cq dk . To prove the lemma, fix one of the vertices at the origin, which we may do since Z d is translation invariant. The number of the remaining k -tuples is ≤ q dk by con-struction. This proves the lemma. IGIDITY, GRAPHS AND HAUSDORFF DIMENSION 27
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