MASS TRANSFERENCE PRINCIPLE: FROM BALLS TO ARBITRARY SHAPES. 1. Introduction. limsupa i = A i.

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1 MASS TRANSFERENCE PRINCIPLE: FROM BALLS TO ARBITRARY SHAPES HENNA KOIVUSALO AND MICHA L RAMS arxiv: v1 [math.ca] 20 Dec 2018 Abstract. The mass transference principle, proved by Beresnevich and Velani 2006, is a strong result that gives lower bounds for the Hausdorff dimension of limsup sets of balls. We present a version for limsup sets of open sets of arbitrary shape, and also calculate the packing dimension of these sets. For (A i ) a sequence of subsets of R d, let 1. Introduction limsupa i = A i. n=1i n Geometry of limsup sets is of great importance in dimension theory, as large classes of fractal sets, including attractors of iterated function systems and random covering sets, are limsup sets. See [AT] and [FJJS] for discussion and references. Our main interest is the following, fundamental result on dimensions of limsup sets, from a 2006 paper of Beresnevitch and Velani [BV]: Theorem 1.1 (Mass transference principle). Let (B i ) be a family of balls in [0,1] d such that λ(limsupb i ) = 1. Let for each i E i be a ball with the same center as B i but of diameter (diamb i ) a,a > 1. Then dim H limsupe i d a. Here λ denotes the Lebesgue measure in R d. This theorem found great many applications in calculating the Hausdorff dimension of limsup sets, in particular in metric number theory. It has also been generalized in several directions. For a recent development see [AB], where a version of this result with different, more general assumptions about B i and limsupb i was shown. Of particular interest for us is the generalisation of Wang, Wu, and Xu [WWX]. In their work, the assumption λ(limsupb i ) = 1, the authors let the sets E i to be not balls of diameter (diamb i ) a but ellipsoids with semiaxes (diamb i ) a,1 a 1... a d. Date: December 21, This proect was supported by OeAD grant number PL03/2017. M.R. was supported by National Science Centre grant 2014/13/B/ST1/01033 (Poland). 1

2 They gave the lower bound min 1 d { d+a } i=1 a i a for the Hausdorff dimension of limsupe i. In the current work, we provide the following new interpretation of this formula: there exists s d depending only on a 1,...,a d such that ϕ s (E i ) λ(b i ), and this s is the lower bound for dim H limsupe i. Here ϕ s is what in the dimension theory of iterated function systems is known as Falconer s singular value function, see [F2] and Section 2. In thisnote we will generalize thisresult to completely general shapes E i : we will only assume that E i B i and that they are open and nonempty. We will not only provide Hausdorff dimension bound for these sets, but also calculate their packing dimension. The argument involves a generalisation of the singular value function, see Section 2. Our results are formulated in Section 3, which is then followed by the proof of the results. 2. Singular value function In 1988 Kenneth Falconer [F2] introduced a function, the singular value function, which for an ellipsoid E R d with semiaxes α 1... α d and parameter s [0,d] assigns the value ϕ s (E) = α 1 α 2...α m α s m m+1, where m = s is the largest integer not larger than s. We will generalize this definition for all nonempty, bounded open sets. Denote by B r (x) the ball of radius r and center x. Let E R d be open and bounded. We define ϕ s (E) = sup µ inf inf x E r>0 r s µ(e B r (x)), where the supremum is taken over Borel probability measures supported on E. On a first glance this definition might seem cumbersome to use, but it turns out that it is enough to consider absolutely continuous measures: Lemma 2.1. There exists κ 1 > 0 such that for every open bounded set E R d there exists a probabilistic absolutely continuous measure η of bounded density, such that the support of η is a finite union of disoint d-dimensional cubes contained in E and ϕ s (E) κ 1 inf inf r s x E r>0 η(e B r (x)). Proof. Fix ε > 0. Let µ 1 be a probability measure supported on E such that ϕ s (E) (1+ε) inf inf r s x E r>0 µ 1 (E B r (x)). For δ > 0 let E δ denote the points in E lying at distance greater than δ from E. We choose δ so small that µ 1 (E δ ) 1 ε and define 2

3 Note µ 2 = 1 µ 1 (E δ ) µ 1 Eδ. r s inf inf (1 ε) inf x E r>0 µ 2 (E B r (x)) inf r s x E r>0 µ 1 (E B r (x)). Let f δ be the normalized characteristic function of B δ (0) and define dµ 3 (x) = f δ (x y)dµ 2 (y), whichisanabsolutelycontinuousprobabilitymeasurewithdensityboundedby(λ(b δ (0))) 1. For x E and r δ we have µ 3 (B r (x)) µ 2 (B r+δ (x)) µ 2 (B 2r (x)). For x E and r < δ we have µ 3 (B r (x)) rd δ dµ 2(B r+δ (x)) rd δ dµ 2(B 2δ (x)). Hence, for every x E and r > 0 one can find r > 0 such that r s µ 3 (B r (x)) (r ) s 2 s µ 2 (B r (x)). Finally, we choose some finite union F E of disoint cubes such that µ 3 (F) 1 ε and define We have inf inf x E r>0 which ends the proof. r s η(e B r (x)) η = 1 µ 3 (F) µ 3 F. (1 ε) inf inf r s x E r>0 µ 3 (E B r (x)) We leave to the reader the exercise of checking that our definition of ϕ s (E) is equivalent (up to a multiplicative constant) to the Falconer s definition if E is an ellipsoid. 3. Statement of results Theorem3.1. Let(B i ) be asequence ofballsin [0,1] d R d such thatλ(limsup i B i ) = 1. Let (E i ) be a sequence of open sets, such that E i B i. Define Then and s = sup{t λ(limsup{b i ϕ t (E i ) λ(b i )}) = 1}. dim H limsupe i s dim P limsupe i = d. 3

4 We will actually prove the following result; it is clear that Theorem 3.1 is an immediate corollary. Theorem3.2. Let(B i ) be asequence ofballsin [0,1] d R d such thatλ(limsup i B i ) = 1. Let (E i ) be a sequence of open sets, such that E i B i. Assume that for some s 0 each pair (B i,e i ) satisfies ϕ s (E i ) λ(b i ). Then dim H limsupe i s and dim P limsupe i = d. Remark 3.3. Noticethat, inparticular, thesets (E i ) being ballsasin[bv]orellipsoids as in[wwx] satisfy the assumptions of Theorem 3.1, so that Theorem 3.1 recovers these dimension results. Furthermore, as is the case for these results as well, the lower bound we provide can be sharp, see e.g. [WWX, Corollary 5.1]. The strategy of the proof of Theorem 3.1 is as follows: We will construct a large Cantor subset C of limsupe i, define a mass distribution µ on the construction tree of C and calculate the local dimension of µ. This will give a lower bound to the dimension. 4. Construction of the Cantor subset We note that we can freely assume that the size of balls B i forms a nonincreasing sequence converging to 0. Indeed, the statement of the theorem does not depend on the order of B i s, and moreover if the size of the balls B i has a non-zero lower bound and if ϕ s (E ni ) > λ(b ni ) for some s > 0 and some subsequence E ni then by the definition of ϕ we will have a nonzero lower bound for λ(e ni ), and hence for λ(limsupe ni ) as well. For a ball B, denote by MB a ball of the same center and M times the radius. The following lemma has been proven as [BV, Lemma 5], but for completeness we will present a proof. Lemma 4.1. Assume λ(limsupb i ) = 1. Then there exists κ 2 > 0 such that for every cube C [0,1] d there exists a finite family of balls B ni C such that the balls 3B ni are pairwise disoint and that λ(bni ) κ 2 λ(c). Proof. Let r denote the side of C. As the diameter of balls B i converges to 0, for any positive ε we know that 3B i B i (1 ε)c, i;b i C i;b i C where (1 ε)c denotes a cube of the same center as C but of side (1 ε)r. Applying the 5r-covering theorem [M, Theorem 2.1], we find a (finite or countable) subfamily of balls B ik C such that 15Bik (1 ε)c, 4

5 and that the balls 3B ik are disoint. Hence, λ(bik ) r d (1 ε) d 15 d and we can choose a finite subfamily such that λ(bik ) r d (1 2ε) d 15 d. As λ(c) = r d, we are done. Before we start the proof: for every set E i denote by η i the absolutely continuous measure provided by Lemma 2.1 and by l i the supremum of its density. We will denote by Ẽi the support of η i. It is enough for us to prove the lower bound for dim H limsupẽi and dim P limsupẽi. We will now inductively construct a family of sets F 0 F 1... such that each F ; 1 is a finite union of some Ẽi s. Clearly, F := F limsupẽi limsupe i. We will then proceed by distributing a measure µ on F. Start with the cube F 0 = [0,1] d. By Lemma 4.1 we can find a finite family of disoint balls F 1 {B i } such that B i F 1 λ(b i ) > κ 2. Let F 1 = B i F 1 Ẽ i. Fix some sequence ε ց 0 and let 1 r 1 = min(r 1,(κ 2 min{ l i λ(b i ) ;B i F 1 }) 1/ε 1 ). Recall that each Ẽi is a finite union of cubes. Now divide all the components of all Ẽ i for B i F 1 into cubes D (1) k of diameter between r 1 /2 and r 1 (notice that different components might need to be divided into cubes of different size) and apply Lemma 4.1 to each of them. We obtain a family F 2 of balls B i such that for each D (1) k, B i F 2 ;B i D (1) k λ(b i ) κ 2 λ(d (1) k ). Let F 2 = B i F 2 Ẽ i. We repeat the construction inductively, defining r as the smallest diameter of components in F and setting 1 r = min(r, r 1 (κ 2 min{ l i λ(b i ) ;B i F }) 1/ε ), and then dividing the set F into cubes D () k of diameter approximately r, applying Lemma 4.1 to each of them and obtaining in that way F +1 and F +1. 5

6 5. Construction of the mass distribution Now we will construct the mass distribution on F. Begin by setting the notations F (E) = {B i F B i E} and F (E) = B i F (E) for E F 0. We start with µ 0 defined as the Lebesgue measure λ restricted to F 0. On the first level of construction, F 1, define ν 1 Bi = µ 0 (Bi ) B k F 1 µ 0 (B k ) for all i such that B i F 1, and no mass elsewhere. Then define, for B i F 1 and Ẽ i B i, the measure µ 1 supported on F 1 by setting µ 1 Ẽi = ν 1 (B i ) η i. Continue inthis way; assume that µ n 1 hasbeen defined on thesets Ẽi with B i F n 1, and let B k F n, B k D (n 1) Ẽi. Then define ν n Bk = µ n 1 (D (n 1) )λ Bk λ(b l ), B l F n(e i );B l D (n 1) Ẽ i and for Ẽk B k F n µ n Ẽk = ν n(b k ) η k, obtaining a measure supported on F n. Notice that (µ n ) is a sequence of probability measures supported on the compact set [0,1] d, so that it has a weakly convergent subsequence. Denote the limit of this subsequencebyµ,andnoticethatitisbyconstructionsupportedonf.infact,µ n (B i ) = µ n+k (B i ) for all k 0, for all B i F n, and similarly for Ẽi B i F n. 6. Calculation of the local dimension Pick a point x F and r > 0. We want to give an estimate to the µ-measure of the ball B r (x). Let n be such that r n < r r n 1. Since x F, we can write x B in B in 1... B i1, with B ik F k for all k. There are two cases to consider: diamb in r < r n 1 and r n r < diamb in. Case 1: diamb in r < r n 1. Recall that in the construction we divide the set Ẽi n 1 into the (n 1)-st generation cubes D (n 1) of diameter approximately r n 1, and each of them has the measure where µ(d (n 1) C n 1 (D (n 1) ) = C n 1 (D (n 1) ) λ(d (n 1) ), ) λ(b i 1 ) λ(b in 1 ) l i1 l in 1. 6 κ2 n 1

7 Let D (n 1) be the (n 1)-st generation cube containing x. We will write C n 1 (x) for C n 1 (D (n 1) ). Recall that r n 1 was chosen in such a way that r n 1 1/ε n 1 log( l i n 1 λ(b in 1 ) )+log r n 2 κ 2 = 1/ε 1 log( l i 1 λ(b i1 ) )+ +1/ε n 1 log( l i n 1λ(B in 1) ). κ 2 κ 2 In particular, we have logc n (y) (6.1) lim max = 0. n y log r n Further, in the cube D (n 1) we find by Lemma 4.1 a collection of balls B i F n such that each of them satisfies ν n (B i ) 1 C n 1 (x)λ(b i ) κ2 and 3B i are disoint. Observe that it is enough to estimate µ(b r (x) D (n 1) ) instead of µ(b r (x)) because, as r < r n 1, the ball B r (x) can intersect at most 5 d n 1-st generation cubes, and if y D (n 1) F B r (x) then B r (x) D (n 1) B 2r (y) D (n 1). We can thus write µ(b r (x) D (n 1) ) µ(b i ). B i F n;b i B r(x) However, by the construction, balls 3B in and 3B i are disoint for any i i n, and in particular x / 3B i. Hence, if B r (x) intersects B i then diamb i r, and we have µ(b r (x) D (n 1) ) λ(b 2r (x)) Cn 1(x). κ 2 Summing up the argument, we get µ(b r (x)) 20 d 1 maxc n 1 (y) y and hence for diamb in r < r n 1 we have (6.2) with q n 0. logµ(b r (x)) logr κ2 d q n Case 2: r n r < diamb in. In this case B r (x) is not going to intersect any B i F n,i i n. Hence, µ(b r (x)) = µ(b r (x) Ẽi n ). Consider the distribution of measure µ on Ẽi n. We have (6.3) µ n Ẽin C n 1 (x)κ 1 2 λ(b in ) η in. Hence, for each of the n-th level cubes D (n) we have µ(d (n) ) = µ n (D (n) ) C n 1 (x)κ 1 2 λ(b i n ) η in (D (n) ). 7

8 We note that these are n-th generation cubes, of size approximately r n, not the (n 1)- st generation cubes we considered in the previous case. However, we do not yet know how exactly µ is distributed on each D (n) this will be decided on the following stages of the construction. Nevertheless, we can write µ(b r (x) Ẽi n ) µ(d (n) ) D (n) ;D (n) B r(x) and we also know that if D (n) B r (x) then D (n) B r+ rn (x). Together with (6.3) we get µ(b r (x)) C n 1 (x)κ 1 2 λ(b in )η in (B r+ rn (x) Ẽi n ). Note that r + r n 2r. By the definition of η in we have hence η in (B 2r (x)) (2r)s κ 1 λ(b in ), logµ(b r (x)) (6.4) slog2+logc n 1(x)+logκ 1 logκ 2 +s s+q n logr logr with q n 0. We finish the proof of Theorem 3.2 applying the mass distribution principle [F, Proposition 2.3] to (6.2) and (6.4). References [AB] D. Allen, S. Baker, A General Mass Transference Principle, preprint 2018, arxiv: [AT] D. Allen, S. Troscheit, The Mass Transference Principle: Ten years on, preprint 2016, arxiv: [BV] V. Beresnevich, S. Velani, A mass transference principle and the Duffin-Schaeffer conecture for Hausdorff measures. Ann. of Math. 164 (2006), , 4 [F] K. J. Falconer, Techniques in Fractal Geometry, John Wiley and Sons, Chichester, [F2] K. J. Falconer, The Hausdorff dimension of self-affine fractals, Math. Proc. Cambridge Philos. Soc. 103 (1988), [FJJS] D.-J. Feng, E. Järvenpää, M. Järvenpää, V. Suomala, Dimensions of random covering sets in Riemann manifolds, Ann. of Prob. 46 (2018), [M] P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge University Press, Cambridge, [WWX] B.-W. Wang, J. Wu, J. Xu, Mass transference principle for limsup sets generated by rectangles, Math. Proc. Cambridge Philos. Soc. 158 (2015), , 4 Henna Koivusalo, University of Vienna, Oskar Morgensternplatz 1, 1090 Vienna, Austria address: henna.koivusalo@univie.ac.at Micha l Rams, Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, Warszawa, Poland address: rams@impan.pl