Simplices in thin subsets of Euclidean spaces

Let $\De$ be a non-degenerate simplex on $k$ vertices. We prove that there exists a threshold $s_k<k$ such that any set $A\subs \R^k$ of Hausdorff dimension $dim\,A\geq s_k$ necessarily contains a similar copy of the simplex $\De$.


Introduction.
A classical problem of geometric Ramsey theory is to show that a sufficiently large sets contain a given geometric configuration.The underlying settings can be the Euclidean space, the integer lattice or vector spaces over finite fields.By a geometric configuration we understand the collection of finite point sets obtained from a given finite set F ⊆ R k via translations, rotations and dilations.
If the size is measured in terms of the positivity of the Lebesgue density, then it is known that large sets in R k contain a translated and rotated copy of all sufficiently large dilates of any non-degenerate simplex ∆ with k vertices [2].However, on the scale of the Hausdorff dimension s < k this question is not very well understood, the only affirmative result in this direction obtained by Iosevich-Liu [6].
In the other direction, a construction due to Keleti [9] shows that there exists set A ⊆ R of full Hausdorff dimension which do not contain any non-trivial 3-term arithmetic progression.In two dimensions an example due to Falconer [3] and Maga [11] shows that there exists set A ⊆ R 2 of Hausdorff dimension 2, which do not contain the vertices of an equilateral triangle, or more generally a non-trivial similar copy of a given non-degenerate triangle.It seems plausible that examples of such sets exist in all dimensions, but this is not currently known.See ( [4]) for related results.
The purpose of this paper is to show that measurable sets A ⊆ R k of sufficiently large Hausdorff dimension s < k contain a similar copy of any given non-degenerate k-simplex with bounded eccentricity.Our arguments make use of and have some similarity to those of Lyall-Magyar [10].We also extend out results to bounded degree distance graphs.For the special case of a path (or chain), and, more generally, a tree, similar but somewhat stronger results were obtained in [1] and [8].

Main results.
Let V = {v 1 , . . ., v k } ⊆ R k be a non-degenerate k-simplex, a set of k vertices which are in general position spanning a k − 1-dimensional affine subspace.For 1 ≤ j ≤ k let r j (V ) be the distance of the vertex v j to the affine subspace spanned by the remaining vertices v i , i = j and define r(V ) := min 1≤j≤k r j (V ).Let d(V ) denote the diameter of the simplex, which is also the maximum distance between two vertices.Then the quantity δ(V ) := r(V )/d(V ), which is positive if and only if V is non-degenerate, measures how close the simplex V is to being degenerate.We say that a simplex Remark 2.1.Note that the dimension condition is sharp for k = 2 as a construction due to Maga [11] shows the existence of a set E ⊆ R 2 with dim(E) = 2 which does not contain any equilateral triangle or more generally a similar copy of any given triangle.
Remark 2.2.It is also interesting to note that the proof of Theorem 1 above proves much more than just the existence of vertices of V ′ similar to V inside E. The proof proceeds by constructing a natural measure on the set of simplexes and proving an upper and a lower bound on this measure.This argument shows that an infinite "statistically" correct "amount" of simplexes V ′ s that satisfy the conclusion of the theorem exist, shedding considerable light on the structure of set of positive upper Lebesgue density.
Remark 2.3.Theorem 1 establishes a non-trivial exponent s 0 < k, but the proof yields s 0 very close to k and not explicitly computable.The analogous results in the finite field setting (see e.g.[5], [7] and the references contained therein) suggest that it may be possible to obtain explicit exponents, but this would require a fundamentally different approach to certain lower bounds obtained in the proof of Theorem 1.
A distance graph is a connected finite graph embedded in Euclidean space, with a set of vertices V = {v 0 , v 1 , . . ., v n } ⊆ R d and a set of edges E ⊆ {(i, j); 0 ≤ i < j ≤ n}.We say that a graph Γ = (V, E) has degree at most k if |V j | ≤ k for all 1 ≤ j ≤ n, where V j = |{v i : (i, j) ∈ E}|.The graph Γ is called proper if the sets V j ∪ {v j } are in general position.Let r(Γ) be the minimum of the distances from the vertices v j to the corresponding affine subspace spanned by the sets V j and note that r(Γ) > 0 if Γ is proper.Let d(Γ) denote length of the longest edge of Γ and let δ(Γ) := r(Γ)/d(Γ).
We say that a distance graph Γ ′ = (V ′ , E) is isometric to Γ, and write Γ ′ ≃ Γ if there is a one-one and onto mapping φ : for all (i, j) ∈ E. One may picture Γ ′ obtained from Γ by a translation followed by rotating the edges around the vertices, if possible.By λ • Γ we mean the dilate of the distance graph Γ by a factor λ > 0 and we say that Note that Theorem 2 implies Theorem 1 as a non-degenerate simplex is a proper distance graph of degree k − 1.

Proof of Theorem 1.
Let E ⊆ B(0, 1) be a compact subset of the unit ball B(0, 1) in R k of Hausdorff dimension s < k.It is well-known that there is a probability measure µ supported on E such that µ(B(x, r)) ≤ C µ r s for all balls B(x, r).The following observation shows that we may take C µ = 4 for our purposes. 1emma 1.There exists a set E ′ ⊆ B(0, 1) of the form E ′ = ρ −1 (F − u) for some ρ > 0, u ∈ R k and F ⊆ E, and a probability measure µ ′ supported on E ′ which satisfies µ ′ (B(x, r) ≤ 4r s , for all x ∈ R k , r > 0. (3.1) Proof.Let K := inf(S), where By Frostman's lemma [?] we have that S = ∅, K > 0, moreover for all balls B(x, r).There exists a ball Q = B(v, ρ) or radius ρ such that µ(Q) ≥ 1 2 Kρ s .We translate E so Q is centered at the origin, set Note that for all balls B = B(x, r), Finally we define the probability measure µ ′ , by µ ′ (A) := µ F (ρA).It is supported on Clearly E contains a similar copy of V if the same holds for E ′ , thus one can pass from E to E ′ and hence assuming that (3.1) holds, in proving our main results.Given ε > 0 let where ψ ≥ 0 is a Schwarz function whose Fourier transform, ψ, is a compactly supported smooth function, satisfying ψ(0) = 1 and 0 ≤ ψ ≤ 1.
We define Let V = {v 0 = 0, . . ., v k−1 } be a given a non-degenerate simplex and note that in proving Theorem 1 we may assume that is a sphere of dimension k − j, of radius r j = r j (V ) ≥ r(V ) > 0. Let σ x 1 ,...,x j−1 denote its normalized surface area measure.
Given 0 < λ, ε ≤ 1 define the multi-linear expression, which may be viewed as a weighted count of the isometric copies of λ∆.
The support of µ ε is not compact, however as it is a rapidly decreasing function it can be made to be supported in small neighborhood of the support of µ without changing our main estimates.Let φ ε (x) := φ(c ε −1/2 x) with some small absolute constant c > 0, where 0 ≤ φ(x) ≤ 1 is a smooth cut-off, which equals to one for |x| ≤ 1/2 and is zero for |x| ≥ 2. Define ψε = ψ ε φ ε and με = µ * ψε .It is easy to see that με ≤ µ ε and με ≥ 1/2, if c > 0 is chosen sufficiently small.Using the trivial upper bound, for it follows that estimate (3.5) remains true with µ ε replaced with με . Let We apply Theorem 2 (ii) together with the more precise lower bound (18) in [10] for the set A ε .This gives that there exists and interval , where for all λ ∈ I, for a constant c = c(k, ψ, r(V )) > 0.
Now, let and in particular 1 0 λ 1/2 T λV (µ) dλ < ∞.On the other hand by (3.8), one has (3.10) Assume that r(V ) ≥ δ, fix a small ε = ε k,δ > 0 and the choose s = s(ε, δ) < k such that thus there exist λ > 0 such that T λV (µ) > 0. Fix such a λ, and assume indirectly that E k = E × . ..×E does not contain any simplex isometric to λV , i.e. any point of the compact configuration space S λV ⊆ R k 2 of such simplices.By compactness, this implies that there is some η > 0 such that the η-neighborhood of E k also does not contain any simplex isometric to λV .As the support of με is contained in the C k ε 1/2 -neighborhood of E, as E = supp µ, it follows that T λV (μ ε ) = 0 for all ε < c k η 2 and hence T λV (µ) = 0, contradicting our choice of λ.This proves Theorem 1.

4.
The configuration space of isometric distance graphs.
Let Γ 0 = (V 0 , E) be a fixed proper distance graph, with vertex set where We call the algebraic set S Γ 0 the configuration space of isometric copies of the Γ 0 .Note that S Γ 0 is the zero set of the family thus it is a special case of the general situation described in Section 5.
If Γ ≃ Γ 0 with vertex set V = {x 0 = 0, x 1 , . . ., x n } is proper then x = (x 1 , . . ., x n ) is a non-singular point of S Γ 0 .Indeed, for a fixed 1 ≤ j ≤ n let Γ j be the distance graph obtained from Γ by removing the vertex x j together with all edges emanating from it.By induction we may assume that x ′ = (x 1 , . . ., x j−1 , x j+1 , . . ., x n ) is a non-singular point i.e the gradient vectors ∇ x ′ f ik (x), (i, k) ∈ E, i = j, k = j are linearly independent.Since Γ is proper the gradient vectors ∇ x j f ij (x) = 2(x i − x j ), (i, j) ∈ E are also linearly independent hence x is a non-singular point.
In fact we have shown that the partition of coordinates x = (y, z) with y = x j and z = x ′ is admissible and hence (6.4) holds.
Let r 0 = r(Γ 0 ) > 0. It is clear that if Γ ≃ Γ 0 and |x j − v j | ≤ η 0 for all 1 ≤ j ≤ n, for a sufficiently small η = η(r 0 ) > 0, then Γ is proper and r(Γ) ≥ r 0 /2.for given 1 ≤ j ≤ n, let X j := {x i ∈ V ; (i, j) ∈ E} and define As explained in Section 6, S X j is a sphere of dimension d − |X j | ≥ 1 with radius r(X j ) ≥ r 0 /2.Let σ X j denote the surface area measure on S X j and write ν X j := φ j σ X j where φ j is a smooth cut-off function supported in an η-neighborhood of v j with φ j (v j ) = 1.Write x = (x 1 , . . ., x n ), φ(x) := n j=1 φ j (x j ), then by (6.4) and (6.5), one has where x ′ = (x 1 , . . ., x j−1 , x j+1 , . . ., x n ) and The constant c j (Γ 0 ) > 0 is the reciprocal of volume of the parallelotope with sides x j − x i , (i, j) ∈ E which is easily shown to be at least c k r k 0 , as the distance of each vertex to the opposite face is at least r 0 /2 on the support of φ.

Proof of Theorem 2.
Let d > k and again, without loss of generality, assume that d(Γ) = 1 and hence δ(Γ) = r(Γ).Given λ, ε > 0 define the multi-linear expression, Given a proper distance graph Γ 0 = (V, E) on |V | = n vertices of degree k < n one has the following upper bound; This implies again that in dimensions d − 1 4n+2 ≤ s ≤ d, there exists the limit T λΓ 0 (µ) := lim ε→0 T λΓ 0 (µ ε ).Also, the lower bound (3.8) holds for distance graphs of degree k, as it was shown for a large class of graphs, the so-called k-degenerate distance graphs, see [10].Thus one may argue exactly as in Section 3, to prove that there exists a λ > 0 for which and Theorem 2 follows from the compactness of the configuration space S λΓ 0 ⊆ R dn .It remains to prove Lemma 4.

6.
Measures on real algebraic sets.
Let F = {f 1 , . . ., f n } be a family of polynomials f i : R d → R. We will describe certain measures supported on the algebraic set A point x ∈ S F is called non-singular if the gradient vectors ∇f 1 (x), . . ., ∇f n (x) are linearly independent, and let S 0 F denote the set of non-singular points.It is well-known and is easy to see, that if S 0 F = ∅ then it is a relative open, dense subset of S F , and moreover it is an F then there exists a set of coordinates, J = {j 1 , . . ., j n }, with 1 ≤ j 1 < . . .< j n ≤ d, such that j F ,J (x) := det ∂f i ∂x j (x) 1≤i≤n,j∈J = 0. (6.2) Accordingly, we will call a set of coordinates J admissible, if (6.2) holds for at least one point x ∈ S 0 F , and will denote by S F ,J the set of such points.For a given set of coordinates x J let ∇ x J f (x) := (∂ x j f (x)) j∈J and note that J is admissible if and only if the gradient vectors are linearly independent at at least one point x ∈ S F .It is clear that, unless S F ,J = ∅, it is a relative open and dense subset of S F and is a also d − n-dimensional sub-manifold, moreover S 0 F is the union of the sets S F ,J for all admissible J.We define a measure, near a point x 0 ∈ S F ,J as follows.For simplicity of notation assume that J = {1, . . ., n} and let Φ(x) := (f 1 , . . ., f n , x n+1 , . . ., x d ).
if and only if Φ(x) = (0, . . ., 0, x n+1 , . . ., x d ) ∈ V .Let I = {n + 1, . . ., d} and write x I := (x n+1 , . . ., x d ).Let Ψ(x I ) = Φ −1 (0, x I ) and in local coordinates x I define the measure ω F via for a continuous function g supported on U .Note that Jac Φ (x) = j F ,J (x), i.e. the Jacobian of the mapping Φ at x ∈ U is equal to the expression given in (6.2), and that the measure dω F is supported on S F .Define the local coordinates y j = f j (x) for 1 ≤ j ≤ n and y j = x j for n < j ≤ d.Then This shows that the measure dω F (given as a differential d − n-form on S F ∩ U ) is independent of the choice of local coordinates x I .Then ω F is defined on S 0 F and moreover the set S 0 F \S F ,J is of measure zero with respect to ω F , as it is a proper analytic subset on R d−n in any other admissible local coordinates.Let x = (z, y) be a partition of coordinates in R d , with y = x J 2 , z = X J 1 , and assume that for i = 1, . . ., m the functions f i depend only on the z-variables.We say that the partition of coordinates is admissible, if there is a point x = (z, y) ∈ S F such that both the gradient vectors ∇ z f 1 (x), . . ., ∇ z f m (x) and the vectors ∇ y f m+1 (x), . . ., ∇ y f n (x) for a linearly independent system.Partition the system and also a set Since ∇ y f i ≡ 0 for 1 ≤ i ≤ m, it follows that the set of coordinates y) as it only involves partial derivatives with respect to the y-variables.Thus we have an analogue of Fubini's theorem, namely  Let T = T X be the inner product matrix with entries t ij := (x−x i )•(x−x j ) for x ∈ S F .Since (x−x i )•(x−x j ) = 1/2(t i +t j −|x i −x j | 2 ) the matrix T is independent of x.We will show that dω F = c T dσ S F where dσ S F denotes the surface area measure on the sphere S F and c T = 2 −m det(T ) −1/2 > 0, i.e for a function g ∈ C 0 (R d ), i.e. the volume of a parallelotope is the square root of the Gram matrix formed by the inner products of its side vectors.

a
i x i , which implies m i=1 a i = 0 and m i=1 a i x i = 0.By replacing the equations |x −x i | 2 = t i with |x − x 1 | 2 − |x − x i | 2 = t 1 − t i , which is of the form x • (x 1 − x i ) = c i , for i = 2, . . ., m, itfollows that S F is the intersection of sphere with an n−1-codimensional affine subspace Y , perpendicular to the affine subspace spanned by the points x i .Thus S F is an m-codimensional sphere of R d if S F has one point x / ∈ span{x 1 , . . ., x m } and all of its points are non-singular.Let x ′ be the orthogonal projection of x to spanX.If y ∈ Y is a point with |y − x ′ | = |x − x ′ | then by the Pythagorean theorem we have that |y − x i | = |x − x i | and hence y ∈ S F .It follows that S F is a sphere centered at x ′ and contained in Y .
S F g(x) dω F (x) = c T S F g(x) dσ S F (x).(6.5)Let x ∈ S F be fixed and let e 1 , . . ., e d be an orthonormal basis so that the tangent space T x S F = Span{e m+1 , . . ., e d } and moreover we have that Span{∇f 1 , . . ., ∇f m } = Span{e 1 , . . ., e m } .Let x 1 , . . ., x n be the corresponding coordinates on R d and note that in these coordinates the surface area measure, as a d − m-form at x, isdσ S F (x) = dx m+1 ∧ . . .∧ dx d .On the other hand, in local coordinates x I = (x m+1 , . . ., x d ), it is easy to see form (6.2)-(6.3)that j F ,J (x) = 2 m vol(x − x 1 , . . ., x − x m ) and hencedω F (x) = 2 −m vol(x − x 1 , . . ., x − x m ) −1 dx m+1 ∧ . . .∧ dx d ,where vol(x − x 1 , . . ., x − x m ) is the volume of the parallelotope with side vectors x − x j .Finally, it is a well-known fact from linear algebra that vol(x − x 1 , . . ., x − x m ) 2 = det (T ), 4))Consider now algebraic sets given as the intersection of spheres.Letx 1 , . .., x m ∈ R d , t 1 , . .., t m > 0 and F = {f 1 , . .., f m } where f i (x) = |x − x i | 2 − t i for i = 1, . .., m.Then S F is the intersection of spheres centered at the points x i of radius r i = t .If the set of points X = {x 1 , . . ., x m } is in general position (i.e they span an m − 1-dimensional affine subspace), then a point x ∈ S F is non-singular if x / ∈ span X, i.e if x cannot be written as linear combination of x 1 , . . ., x m .Indeed, since ∇f i (x) = 2(x − x i ) we have that i