Low dimensional pinned distance sets via spherical averages
aa r X i v : . [ m a t h . C A ] J a n LOW DIMENSIONAL PINNED DISTANCE SETS VIASPHERICAL AVERAGES
TERENCE L. J. HARRIS
Abstract.
An inequality is derived for the average t -energy of pinned distancemeasures, where 0 < t <
1. This refines Mattila’s theorem on distance sets [8]to pinned distance sets, and gives an analogue of Liu’s theorem [6] for pinneddistance sets of dimension smaller than 1. Introduction
For n ≥ d : R n × R n → [0 , ∞ ) be the Euclidean distance function ( x, y ) x − y | , and for fixed x ∈ R n let d x : R n → [0 , ∞ ) be the pinned distance function d x ( y ) = d ( x, y ). For finite compactly supported Borel measures µ and ν on R n ,define ∆( µ, ν ) by Z φ ( s ) d ∆( µ, ν )( s ) = Z s − ( n − / φ ( s ) d ( µ, ν )( s )= Z Z | x − y | − ( n − / φ ( | x − y | ) dµ ( x ) dν ( y ) , for all non-negative Borel measurable φ : R → [0 , ∞ ]. Let σ ( µ, ν )( r ) = Z S n − \ µ ( rξ ) [ ν ( rξ ) dσ ( ξ ) , r ≥ , where σ is the surface measure on the sphere S n − , and b µ ( ξ ) = Z e − πi h x,ξ i dµ ( x ) . The aim of this work is to give a sufficient condition for the inequality(1.1) Z I t (∆( ν, δ x )) dµ ( x ) . t,α,γ c α ( µ ) I γ ( ν ) , in terms of t , α and γ , via the L spherical averages of b µ . Here c α ( µ ) = sup x ∈ R n r> µ ( B ( x, r )) r α , the mutual t -energy of µ and ν is defined by I t ( µ, ν ) = Z Z | x − y | − t dµ ( x ) dν ( y ) , and I t ( µ ) = I t ( µ, µ ). The main result, Theorem 2.1, refines Mattila’s theorem (see[8, Theorem 4.16 and Theorem 4.17] or [9, Proposition 15.2]) to pinned distancemeasures, and is the Hausdorff dimension analogue to Liu’s theorem [6], whichgives a sufficient condition for the support of some d x ( µ ) to have positive Lebesguemeasure. The proof consists of augmenting Mattila’s proof with a “Cauchy-Schwarz reversal” technique (also used in [6]). I do not know if the proof from [6] can bedirectly extended to lower dimensional pinned distance sets. The two main toolsused in [6] are Lemma 1.1 (stated below) and Liu’s identity [6, Theorem 1.9]:(1.2) Z ∞ | ( σ t ∗ f ) ( x ) | t n − dt = Z ∞ | ( c σ r ∗ f ) ( x ) | r n − dr, x ∈ R n , for Schwartz f on R n , where σ r is the pushforward of σ under the map ξ rξ .One key part of the proof of Mattila’s theorem is the identity [8, Theorem 4.6](1.3) Z ∞ ∆( f, g )( t )∆( h, k )( t ) dt = Z ∞ Σ( f, g )( r )Σ( h, k )( r ) dr, for Schwartz f, g, h, k on R n , whereΣ( f, g )( r ) = r ( n − / σ ( f, g )( r ) , r ≥ . The identity (1.2) can be obtained from (1.3) by formally setting h = f and g = k = δ x , whilst in the original application Mattila used g = f and h = k = f . Thissuggests that using g = k = δ x in Mattila’s original proof should result in a pinneddistance version, and this is the main idea guiding the proof of Theorem 2.1 below.The case µ = ν in (1.1) is related to the distance set problem, which asks whetherthe condition dim A > n/ A ⊆ R n implies that d ( A × A ) has positive Lebesguemeasure. This is still open, as is the stronger pinned version, which asks whetherdim A > n/ x ∈ A such that d x ( A ) has positive measure.Throughout, given α ∈ [0 , n ] let β (cid:0) α, S n − (cid:1) be the supremum over all β ≥ µ with supp µ ⊆ B (0 , Z | b µ ( rξ ) | dσ ( ξ ) . β r − β I α ( µ ) , ∀ r > . The following “Cauchy-Schwarz reversal” lemma will be needed. In [9, p. 197] it isattributed to [1, Lemma C.1]; the version below is identical to Lemma 3.2 from [6].
Lemma 1.1.
Let µ be a Borel measure supported in the unit ball of R n . Then forany α ∈ [0 , n ] and any ǫ > , Z | ( c σ r ∗ f ) ( x ) | dµ ( x ) . α,ǫ c α ( µ ) r ǫ − β ( α,S n − ) Z (cid:12)(cid:12)(cid:12) b f ( rξ ) (cid:12)(cid:12)(cid:12) dσ ( ξ ) , ∀ r ≥ , for all Schwartz f . Average energies of pinned distance measures
The following theorem is the main result.
Theorem 2.1.
Let n ≥ . If α, γ ∈ [0 , n ] and (2.1) 0 < t < γ + β (cid:0) α, S n − (cid:1) − n + 1 ≤ , then for any Borel measures µ and ν supported in the unit ball of R n , (2.2) Z I t ( d ( ν, δ x )) dµ ( x ) . t,α,γ c α ( µ ) I γ ( ν ) , γ ≤ ( n − / , Z I t (∆( ν, δ x )) dµ ( x ) . t,α,γ c α ( µ ) I γ ( ν ) , γ > ( n − / . OW DIMENSIONAL PINNED DISTANCE SETS 3
One corollary of Theorem 2.1 is that for any Borel set A with ( n − / < dim A ≤ n/ ∀ ǫ > ∃ x ∈ A : dim d x ( A ) ≥ dim A − ( n − − ǫ. This is originally due to Oberlin-Oberlin [10], and follows from (2.2), Frostman’slemma and the lower bound β ( α, S n − ) ≥ ( n − / α ≥ ( n − / n ∈ { , } with dim A close to n/
2, Shmerkin’s bound from [12] gives animprovement over (2.3) by a small absolute constant, and therefore (2.3) is notsharp whenever dim A is sufficiently close to n/ n ∈ { , } .For dim A > n/
2, another corollary of Theorem 2.1 is that(2.4) ∀ ǫ > ∃ x ∈ A : dim d x ( A ) ≥ min (cid:26) , A − dim An − ( n − (cid:27) − ǫ, which follows from (2.2) and the following inequality from [4]: β (cid:0) α, S n − (cid:1) ≥ α ( n − n , n/ ≤ α ≤ n. When n = 2, (2.4) is weaker than the combined results of [7, 11, 12] for all valuesof dim A . For all even n ≥
4, (2.4) is weaker than what would likely follow fromthe methods in [3, 7]. For odd n ≥
5, and n < dim A ≤ n n − , (2.4) is new (as far as I am aware). A similar statement holds if n = 3, althoughfor dim A close to 3 / n/ A is sufficiently closeto 3 / Proof of Theorem 2.1.
Assume that γ > ( n − /
2; the proof of the case γ ≤ ( n − / µ and ν are probability measures. Let f and g be non-negative smooth compactlysupported functions on R n , and abbreviate ∆( f, g ) = ∆, which is a finite measureby [8, Lemma 4.3]. Then(2.5) I t (∆) . | I t ( F ) | + | I t (∆ , K ) | ([8, p. 221]) . The functions F , K and corresponding quantities I t ( F ), I t (∆ , K ) from [8] will notbe redefined here; all that will be needed is that F satisfies | I t ( F ) | . Z ∞ r t + n − | σ ( f, g )( r ) | dr ([8, p. 221])(2.6) . k f k k g k + Z ∞ r t + n − | σ ( f, g )( r ) | dr, and that K satisfies both(2.7) | I t (∆ , K ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ Z ∞ ∆( s ) K ( x ) | s − x | − t ds dx (cid:12)(cid:12)(cid:12)(cid:12) ([8, p. 221]) , and(2.8) K ( x ) = x / Z ∞ r n/ R ( rx ) σ ( f, g )( r ) dr, x > , ([8, Lemma 4.14]) , T. L. J. HARRIS where R : (0 , ∞ ) → C is some Borel function with(2.9) | R ( x ) | . min n x − / , x − / o , ∀ x > | R ( x ) | . x ( t − / , ∀ x > . Using (2.7), (2.8) and then (2.10) gives | I t (∆ , K ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ ∆( s ) Z ∞ r n/ σ ( f, g )( r ) Z ∞ | s − x | − t x / R ( rx ) dx dr ds (cid:12)(cid:12)(cid:12)(cid:12) . Z ∞ ∆( s ) Z ∞ r ( n + t − / | σ ( f, g )( r ) | Z ∞ | s − x | − t x ( t − / dx dr ds ∼ (cid:18)Z ∞ s − t/ ∆( s ) ds (cid:19) · (cid:18)Z ∞ r ( n + t − / | σ ( f, g )( r ) | dr (cid:19) (2.11) . (cid:18)Z ∞ r ( n + t − / | σ ( f, g )( r ) | dr (cid:19) (2.12) . ǫ k f k k g k + Z ∞ r t + n + ǫ − | σ ( f, g )( r ) | dr. (2.13)To get from (2.11) to (2.12), the term R ∞ s − t/ ∆( s ) ds is equal to I ( n + t − / ( f, g )by definition, which equals a constant multiple of R ∞ r ( n + t − / σ ( f, g )( r ) dr bypolar coordinates and the Fourier formula for the mutual energy [8, Eq. 3.5].Substituting (2.6) and (2.13) into (2.5) gives(2.14) I t (∆( f, g )) . ǫ k f k k g k + Z ∞ r t + n + ǫ − | σ ( f, g )( r ) | dr, for any smooth, non-negative compactly supported functions f and g . For eachinteger j ≥ φ j ( z ) = j n φ ( jz ), where φ is a non-negative radial bump functionon R n which is compactly supported in the unit ball and satisfies R φ = 1. Forfixed x ∈ supp µ , taking f = f j = ν ∗ φ j and g = g j,x = δ x ∗ φ j in (2.14) gives(2.15) I t (∆( ν, δ x )) ≤ lim inf j →∞ I t (∆ ( f j , g j,x )) . ǫ j →∞ Z ∞ r t + n + ǫ − | σ ( f j , g j,x ) ( r ) | dr. The first inequality in (2.15) is justified by the following argument. If ν m := ν ↾ A m where A m = n y ∈ supp ν : R | y − z | − γ dν ( z ) ≤ m o , then c γ ( ν m ) < ∞ (see [9,p. 20]), so by using I t (∆( ν, δ x )) = lim m →∞ I t (∆( ν m , δ x )) , I t (∆( ν m ∗ φ j , g j,x )) ≤ I t (∆( f j , g j,x )) , it may be assumed in proving the first part of (2.15) that c γ ( ν ) < ∞ . By [2,Theorem 2.8] f j × g j,x ∗ ⇀ ν × δ x , and hence d ( f j × g j,x ) ∗ ⇀ d ( ν × δ x ). Using thedefinition of ∆, Fubini’s theorem, and the assumption c γ ( ν ) < ∞ gives(2.16) ∆ ( f j × g j,x ) [0 , ε ] . c γ ( ν ) ε γ − ( n − / , ∀ ε > , and similarly(2.17) ∆ ( ν × δ x ) [0 , ε ] . c γ ( ν ) ε γ − ( n − / , ∀ ε > . OW DIMENSIONAL PINNED DISTANCE SETS 5
The condition d ( f j × g j,x ) ∗ ⇀ d ( ν × δ x ) combined with (2.16) and (2.17) yields∆( f j × g j,x ) ∗ ⇀ ∆( ν × δ x ). Therefore the Fourier transform of ∆( f j , g j,x ) convergespointwise to the Fourier transform of ∆( ν, δ x ). Fatou’s lemma and the Fourierformula for energy ([8, Eq. 3.5]) then give the first part of (2.15).Integrating the outer parts of (2.15) with respect to µ gives(2.18) Z I t (∆( ν, δ x )) dµ ( x ) . ǫ j →∞ Z ∞ Z r t + n + ǫ − | σ ( f j , g j,x ) ( r ) | dµ ( x ) dr. The inequality | σ ( f j , g j,x ) ( r ) | ≤ | σ ( ν, δ x ) ( r ) | holds for every r > x ∈ supp µ ,since φ is a radial bump function which integrates to 1. Moreover, σ ( f j , g j ) ( r ) → σ ( ν, δ x )( r ) as j → ∞ , pointwise for every r > x ∈ supp µ . Therefore applyingdominated convergence and then Lemma 1.1 to (2.18) gives Z I t (∆( ν, δ x )) dµ ( x ) . ǫ Z ∞ Z r t + n + ǫ − | σ ( ν, δ x ) ( r ) | dµ ( x ) dr = 1 + Z ∞ r t + n + ǫ − Z | ( c σ r ∗ ν ) ( x ) | dµ ( x ) dr . ǫ c α ( µ ) Z ∞ r t + n − ǫ − β ( α,S n − ) Z | b ν ( rξ ) | dσ ( ξ ) dr . c α ( µ ) I γ ( ν ) , by (2.1), provided ǫ is small enough, again using polar coordinates and the Fourierformula for the energy [8, Eq. 3.5]. This proves (2.2) when γ > ( n − /
2. The onlyadjustment required in the case γ ≤ ( n − / I t ( d ( ν, δ x )) . lim inf j →∞ I t (∆ ( f j , g j,x )). (cid:3) References [1] Barcelo, J. A., Bennett, J. M., Carbery, A., Rogers, K., M.: On the dimension of divergencesets of dispersive equations. Math. Annalen L estimates of the Schr¨odinger maximal function in higher dimen-sions. Ann. of Math. , 837–861 (2019)[5] Falconer, K. J.: On the Hausdorff dimensions of distance sets. Mathematika , 206–212(1986)[6] Liu, B: An L -identity and pinned distance problem. Geom. Funct. Anal. L -method. Proc. Amer. Math. Soc . Cambridge University Press, Cambridge (2015)[10] Oberlin, D., Oberlin, R.: Spherical means and pinned distance sets. Commun. KoreanMath. Soc. Department of Mathematics, Cornell University, Ithaca, NY 14853-4201, USA
Email address ::