aa r X i v : . [ m a t h . D S ] F e b New dimension bounds for αβ sets Simon Baker
School of Mathematics,University of Birmingham,Birmingham, B15 2TT, UK.
Email: [email protected] 12, 2021
Abstract
In this paper we obtain new lower bounds for the upper box dimension of αβ sets. As a corollary of our main result, we show that if α is not a Liouville num-ber and β is a Liouville number, then the upper box dimension of any αβ set is 1. Wealso use our dimension bounds to obtain new results on affine embeddings of self-similar sets. Mathematics Subject Classification 2010 : 11J04, 11K06, 28A80.
Key words and phrases : αβ sets, Diophantine approximation, Self-similar sets. Let T := R / Z denote the unit circle. Given α, β ∈ R \ Q , a non-empty closed set E ⊂ T iscalled an αβ set if for all x ∈ E either x + α mod 1 ∈ E or x + β mod 1 ∈ E . A sequence( x n ) n ≥ of points in T is called an αβ orbit if for all n ≥ , either x n +1 − x n = α mod 1 or x n +1 − x n = β mod 1 . Clearly any αβ set contains an αβ orbit. If α and β are rationallydependent modulo one, i.e. there exists n , n ∈ Z such that n α + n β = 0 mod 1 , thenusing the well known fact that orbits of irrational circle rotations are dense in T together withthe Baire category theorem, it can be shown that every αβ set has non-empty interior (see [9,Theorem 1.5(i)]). This observation naturally leads to the following question that was posed byEngelking in [6]: Suppose that α and β are rationally independent modulo one, do there existnowhere dense αβ sets? This question was answered by Katznelson in [11]. He proved that if α and β are rationally independent, then there do exist nowhere dense αβ sets. Katznelson alsoproved that αβ sets exist with arbitrarily small Hausdorff dimension. Interest in αβ sets wasrenewed in a recent paper of Feng and Xiong [9]. In this paper they connected αβ sets and theirhigher dimensional analogues to the existence of affine embeddings of self-similar sets. Theyproved that if α and β are rationally independent then any αβ set E satisfies E − E = T or E has non-empty interior. This result implies that if α and β are rationally independent thenany αβ set E satisfies dim B E ≥ /
2. Further results on the dimension of αβ sets and theirhigher dimensional analogues were obtained by Yu in [14]. In this paper Yu conjectured that forrationally independent α and β, any αβ set E satisfies dim B E = 1 . In this paper we obtain Instead of just considering two elements α, β ∈ R \ Q , one can consider α , . . . , α n ∈ R \ Q and then defineappropriate analogues of αβ sets and αβ orbits. This conjecture was formulated in [14] in terms of the lower box dimension. Our formulation is easily seen tobe equivalent. αβ sets. These bounds depend upon theDiophantine properties of α and β . As a corollary of our main result, we give the first examplesof α and β satisfying the conclusion of Yu’s conjecture where box dimension is replaced withupper box dimension. We conclude this introductory section by mentioning a paper of Chen,Wang, and Wen [5] who considered random analogues of αβ orbits. They proved that suchsequences were almost surely uniformly distributed modulo one, and that the exponential sumsalong the orbit have square root cancellation. A well known theorem due to Dirichlet states that for any x ∈ R and Q > , there exists integers p and q such that 1 ≤ q ≤ Q and (cid:12)(cid:12)(cid:12)(cid:12) x − pq (cid:12)(cid:12)(cid:12)(cid:12) < qQ . This implies that if x is an irrational number, then the inequality (cid:12)(cid:12)(cid:12)(cid:12) x − pq (cid:12)(cid:12)(cid:12)(cid:12) < q has infinitely many solutions in integers p and q . Given τ ≥ x ∈ R \ Q is τ -wellapproximable if there exists infinitely many ( p, q ) ∈ Z × N satisfying (cid:12)(cid:12)(cid:12)(cid:12) x − pq (cid:12)(cid:12)(cid:12)(cid:12) < q τ . We denote the set of τ -well approximable numbers by W ( τ ). For x ∈ R \ Q we define the exactorder of x to be τ ( x ) := sup { τ : x ∈ W ( τ ) } . If τ ( x ) = ∞ then we say that x is a Liouville number. For τ ∈ [2 , ∞ ) ∪ {∞} we denote the setof real numbers with exact order τ by E ( τ ). Equipped with these definitions we are now ableto state the main result of this paper. Theorem 1.1.
Let τ , τ ≥ satisfy τ < τ + 2 and suppose that α ∈ E ( τ ) and β ∈ W ( τ ) . Then any αβ orbit ( x n ) n ≥ satisfies dim B ( { x n } ) ≥ − τ − τ . Theorem 1.1 immediately implies the following result.
Corollary 1.2.
Assume that α is not a Liouville number and β is a Liouville number. Thenany αβ orbit ( x n ) n ≥ satisfies dim B ( { x n } ) = 1 . Since every αβ set contains an αβ orbit, we immediately see that suitable analogues ofTheorem 1.1 and Corollary 1.2 also hold for αβ sets. We emphasise that the α and β appearingin the statements of Theorem 1.1 and Corollary 1.2 are rationally independent. This is becauseany rationally dependent α and β must have the same exact order.The rest of this paper is structured as follows. In Section 2 the relevant definitions fromFractal Geometry are given and we gather some useful results from the theory of continuedfractions. In Section 3 we prove Theorem 1.1. In Section 4 we apply Theorem 1.1 to obtain aresult on affine embeddings of self-similar sets. 2 Preliminaries
Let F ⊂ R n and s ≥
0. Given δ > H sδ ( F ) := inf ( ∞ X i =1 Diam ( U i ) s : { U i } is a δ -cover of F ) . We define the s -dimensional Hausdorff measure of F to be H s ( F ) := lim δ → H sδ ( F ) . The Hausdorff dimension of F is given bydim H ( F ) := inf { s ≥ H s ( F ) = 0 } = sup { s ≥ H s ( F ) = ∞} . Given a bounded set F ⊂ R n , we let N ( F, r ) denote the minimum number of closed balls ofradius r required to cover F . The upper box dimension of a bounded set F is defined to bedim B ( F ) := lim sup r → log N ( F, r ) − log r . The lower box dimension is defined similarly using liminf instead of limsup. When the lower andupper box dimensions coincide we refer to the common value as the box dimension and denoteit by dim B ( F ). For more on dimension theory and fractal sets we refer the reader to [7]. Proofs of the properties stated below can be found in the books [3] and [4].For any x ∈ [0 , \ Q , there exists a unique sequence ( a n ) n ≥ ∈ N N such that x = 1 a + 1 a + 1 a + · · · . We call the sequence ( a n ) the continued fraction expansion of x . Suppose x has continuedfraction expansion ( a n ) , then for each n ≥ p n q n := 1 a + 1 a + 1 a + · · · a n . The fraction p n /q n is called the n -th partial quotient of x . For any x ∈ [0 , \ Q , its sequenceof partial quotients satisfies the following properties: • If we set p − = 1 , q − = 0 , p = 0 , q = 1, then for any n ≥ p n = a n p n − + p n − (2.1) q n = a n q n − + q n − . For any n ≥ q n ( q n +1 + q n ) < (cid:12)(cid:12)(cid:12)(cid:12) x − p n q n (cid:12)(cid:12)(cid:12)(cid:12) < q n q n +1 . (2.2) • If q < q n +1 then | qx − p | ≥ | q n x − p n | (2.3)for any p ∈ Z .For x ∈ R we will on occasion use k x k to denote the distance from x to the nearest integer.We will use the following lemma in our proof of Theorem 1.1. Lemma 2.1.
Let x ∈ E ( τ ) for some τ ≥ . Then for any ǫ > , for all q ∈ R sufficiently largethe interval [ q, q τ + ǫ − ] contains the denominator of some partial quotient of x .Proof. Let ( q n ) ∞ n =1 denote the sequence of denominators of partial quotients of x written inincreasing order. Suppose q > q is such that the interval [ q, q τ + ǫ − ] does not contain thedenominator of a partial quotient of x . Then let n ∗ ≥ q n ∗ < q and q n ∗ +1 > q τ + ǫ − . (2.4)Equation (2.1) implies that q n +1 ≤ a n +1 q n (2.5)for all n ≥
1. Combining (2.4) and (2.5) we have2 a n ∗ +1 ≥ q n ∗ +1 q n ∗ > q τ + ǫ − > q τ + ǫ − n ∗ . (2.6)Equations (2.1) and (2.2) imply that (cid:12)(cid:12)(cid:12)(cid:12) x − p n q n (cid:12)(cid:12)(cid:12)(cid:12) ≤ a n +1 q n (2.7)for all n ≥
1. It now follows from (2.6) and (2.7) that (cid:12)(cid:12)(cid:12)(cid:12) x − p n ∗ q n ∗ (cid:12)(cid:12)(cid:12)(cid:12) ≤ q τ + ǫn ∗ . (2.8)Since x ∈ E ( τ ) inequality (2.8) can have only finitely many solutions. It follows that for all q ∈ R sufficiently large the interval [ q, q τ + ǫ − ] must contain the denominator of a partial quotientof x . Let α, β ∈ R \ Q . To any αβ orbit ( x n ) n ≥ we can associate a unique sequence ω = ( ω n ) n ≥ ∈{ α, β } N such that x n − x n − = ω n mod 1for all n ≥ . Given ω ∈ { α, β } N and N ∈ N we let | ω | α,N := { ≤ n ≤ N : ω n = α } and | ω | β,N := { ≤ n ≤ N : ω n = β } . The following proposition shows that if an αβ orbit ( x n ) n ≥ is such that the quantities | ω | α,N and | ω | β,N are not uniformly comparable then { x n } n ≥ is dense in T . roposition 3.1. Let α, β ∈ R \ Q and ( x n ) n ≥ be an αβ orbit. Suppose that for any C > there exists infinitely many N ∈ N such that either | ω | α,N ≥ C · | ω | β,N or | ω | β,N ≥ C · | ω | α,N . Then { x n } is dense in T .Proof. It follows from our hypothesis that the sequence ω must either contain arbitrarily longstrings of consecutive α terms or consecutive β terms. Since both α and β are irrational, andany orbit of an irrational rotation is dense in T , it follows that { x n } must also be dense in T . Proposition 3.2.
Let τ , τ ≥ satisfy τ < τ + 2 and suppose that α ∈ E ( τ ) and β ∈ W ( τ ) . Let ( x n ) n ≥ be an αβ orbit for which there exists C > such that for all N ∈ N sufficientlylarge we have | ω | β,N C ≤ | ω | α,N ≤ C · | ω | β,N . Then dim B ( { x n } ) ≥ − τ − τ . Proof.
Without loss of generality we may assume that α, β ∈ [0 , x n ) n ≥ an αβ orbit satisfying our hypothesis and let ω be the associated unique element of { α, β } N . Without loss of generality we may further assume that x = 0 . This means that forany N ≥ x N = α · | ω | α,N + β · | ω | β,N mod 1 . Notice that | ω | α,N + | ω | β,N = N for all N ≥
1. It follows from this observation and our hypothesisthat there exists
C > , not necessarily the same C as in the statement of our proposition, suchthat NC ≤ | ω | α,N (3.1)for all N sufficiently large.Let ǫ > β ∈ W ( τ ) there exists a sequence of reduced fractions ( p l /q l ) l ≥ such that (cid:12)(cid:12)(cid:12)(cid:12) β − p l q l (cid:12)(cid:12)(cid:12)(cid:12) ≤ q τ l (3.2)for all l ≥
1. Without loss of generality we may assume that the sequence ( q l ) ∞ l =1 is strictlyincreasing. By Lemma 2.1, for all l sufficiently large, there exists q ′ l the denominator of somepartial quotient of α which satisfies q ′ l ∈ (cid:20) q τ − ǫ τ ǫ − l , q τ − ǫ l (cid:21) . For any j ∈ N we let k j denote the minimum of those k ∈ N satisfying αj + βk mod 1 ∈ { x n } . Equivalently k j is the smallest integer such that | ω | α,j + k j = j . Notice that for any N ∈ N , if1 ≤ j ≤ | ω | α,N then we must have k j < N. For all l sufficiently large so that q ′ l is well defined,we let W ( l, p ) := { ≤ j ≤ | ω | α,q ′ l : k j = p mod q l } ≤ p ≤ q l −
1. By the pigeonhole principle and (3.1), for all l sufficiently large thereexists 0 ≤ p ′ ≤ q l − W ( l, p ′ ) ≥ q ′ l Cq l . (3.3)We now set out to prove that the elements of { x n } corresponding to the elements of W ( l, p ′ ) arewell separated. Observe now that for any distinct j, j ′ ∈ W ( l, p ′ ) we have k ( αj + βk j ) − ( αj ′ + βk j ′ ) k ≥ k α ( j − j ′ ) k | {z } (1) − k β ( k j − k j ′ k | {z } (2) . (3.4)We now show how (1) can be bounded from below and (2) can be bounded from above. Noticethat j − j ′ is a non-zero integer satisfying | j − j ′ | < q ′ l . Combining (2.2) and (2.3) it follows that k α ( j − j ′ ) k ≥ q ′ l . (3.5)Now focusing on (2) , let d j , d j ′ ∈ N be such that k j = d j q l + p ′ and k j ′ = d j ′ q l + p ′ . Then wehave k β ( k j − k j ′ ) k ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) β − p l q l (cid:19) ( k j − k j ′ ) (cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13) p l q l ( k j − k j ′ ) (cid:13)(cid:13)(cid:13)(cid:13) ≤ q ′ l q τ l + (cid:13)(cid:13)(cid:13)(cid:13) p l q l ( d j q l − d j ′ q l ) (cid:13)(cid:13)(cid:13)(cid:13) = q ′ l q τ l + k p l ( d j − d j ′ ) k = q ′ l q τ l ≤ q τ / l . (3.6)In the second line in the above we have used (3.2) and the inequality | k j − k j ′ | < q l ′ . Thisinequality follows from the fact that k j and k j ′ are integers satisfying 0 ≤ k j , k j ′ < q l ′ . In thefinal line we used that q ′ l ≤ q τ − ǫ l . Substituting (3.5) and (3.6) into (3.4) we have k ( αj + βk j ) − ( αj ′ + βk j ′ ) k ≥ q ′ l − q τ / l . (3.7)Since q ′ l ≤ q τ − ǫ l , for l sufficiently large we have12 q ′ l − q τ / l ≥ q ′ l − q ′ l q τ / l ! ≥ q ′ l (cid:18) − q ǫl (cid:19) ≥ q ′ l . Using this lower bound in (3.7), it follows that for l sufficiently large, for any distinct j, j ′ ∈ W ( l, p ′ ) we have (cid:13)(cid:13) ( αj + βk j ) − ( αj ′ + βk j ′ ) (cid:13)(cid:13) ≥ q ′ l . Therefore for any l sufficiently large we require at least W ( l, p ′ ) closed balls of radius (10 q ′ l ) − to cover { x n } . Using the lower bound for W ( l, p ′ ) provided by (3.3) and the inequality q ′ l ≥ q τ − ǫ τ ǫ − l , we have dim B ( { x n } ) = lim sup r → log N ( { x n } , r ) − log r ≥ lim sup l →∞ log q ′ l /Cq l log 10 q ′ l − lim inf l →∞ log q l log q l ′ ≥ − τ + ǫ − τ − ǫ . Since ǫ was arbitrary we may concludedim B ( { x n } ) ≥ − τ − τ . Since any dense subset of T has upper box dimension 1, Propositions 3.1 and 3.2 togetherimply Theorem 1.1. We call a map ϕ : R d → R d a similarity if there exists r ∈ (0 , , t ∈ R d , and a d × d orthogonalmatrix O such that ϕ = r · O + t. For our purposes, we call a finite set of similarities Φ = { ϕ i } i ∈ I an iterated function system or IFS for short. A well known result due to Hutchinson [10] statesthat for any IFS Φ , there exists a unique non-empty compact set F ⊂ R d satisfying F = [ i ∈ I ϕ i ( F ) . We call F the self-similar set of Φ. Many well known fractal sets, such as the middle thirdCantor set and the von-Koch curve, can be realised as self-similar sets for appropriate choicesof IFS. If ϕ i ( F ) ∩ ϕ j ( F ) = ∅ for all i = j then we say that Φ satisfies the strong separationcondition. We say that Φ satisfies the open set condition if there exists a non-empty boundedopen O ⊂ R d such that ϕ i ( O ) ⊂ O for all i ∈ I and ϕ i ( O ) ∩ ϕ j ( O ) = ∅ for all i = j .Let A, B ⊂ R d . We say that A can be affinely embedded into B if there exists a map f : R d → R d of the form f ( x ) = M x + a for some invertible matrix M and a ∈ R d whichsatisfies f ( A ) ⊂ B . It is an interesting problem to determine when one self-similar set can beaffinely embedded inside of another. This problem was first studied in [8]. It is reasonable toexpect that if a self-similar set can be affinely embedded inside of another self-similar set whichis totally disconnected, then the underlying contraction ratios should exhibit some arithmeticdependence. With this in mind the authors of [8] formulated the following conjecture. Conjecture 4.1.
Suppose that
E, F are two totally disconnected non-trivial self-similar setsin R d , generated by IFSs Φ = { ϕ i } i ∈ I and Ψ = { ψ j } j ∈ J respectively. Let r i , r ′ j denote thecontraction ratios of ϕ i and ψ j respectively. Suppose that F can be affinely embedded into E .Then for each j ∈ J there exists non-negative rational numbers t i,j such that r ′ j = Q i ∈ I r t i,j i . Inparticular, if r i = r for all i ∈ I, then log r ′ j / log r ∈ Q for all j ∈ J .Conjecture 4.1 was studied in [1, 2, 8, 9, 12, 13]. In [8] it was shown that Conjecture 4.1 istrue if we also assume that Φ satisfies the strong separation condition, r i = r for all i ∈ I, anddim H ( E ) < /
2. Similar results were obtained in [9] without the assumption r i = r for all i ∈ I .These results come at the cost that dim H ( E ) is required to satisfy a stricter upper bound. Inparticular, the results of [9] imply that when Φ consists of two similarities then Conjecture 4.1is true if we also assume that Φ satisfies the strong separation condition and dim H ( E ) < / d = 1. Shmerkin in [12] showed thatConjecture 4.1 is true under the additional assumptions that d = 1, Φ satisfies the open setcondition, r i = r for all i ∈ I, and dim H ( E ) <
1. Wu in [13] obtained the same result asShmerkin but required the stronger assumption that Φ satisfies the strong separation condition.Our main result in this direction is the following theorem.7 heorem 4.2.
Let
Φ = { ϕ i } i ∈ I and Ψ = { ψ j } j ∈ J be two IFSs satisfying the following properties:1. Φ satisfies the strong separation condition.2. There exists r , r ∈ (0 , and I , I ⊂ I such that Φ = { ϕ i, = r O i, + t i, } i ∈ I ∪ { ϕ i, = r O i, + t i, } i ∈ I .
3. There exists j ∗ ∈ J such that:(a) ψ j ∗ = r ′ j ∗ I d + t j ∗ . (b) There exists τ , τ ≥ satisfying τ < τ + 2 and − log r log r ′ j ∗ ∈ E ( τ ) and − log r log r ′ j ∗ ∈ W ( τ ) . Then if dim H ( E ) < (cid:16) − τ − τ (cid:17) then F cannot be affinely embedded into E . Theorem 4.2 has the following corollary.
Corollary 4.3.
Let
Φ = { ϕ i } i ∈ I and Ψ = { ψ j } j ∈ J be two IFSs satisfying the following proper-ties:1. Φ satisfies the strong separation condition.2. There exists r , r ∈ (0 , and I , I ⊂ I such that Φ = { ϕ i, = r O i, + t i, } i ∈ I ∪ { ϕ i, = r O i, + t i, } i ∈ I .
3. There exists j ∗ ∈ J such that:(a) ψ j ∗ = r j ∗ I d + t j ∗ . (b) − log r log r ′ j ∗ is not a Liouville number and − log r log r ′ j ∗ is a Liouville number.Then if dim H ( E ) < then F cannot be affinely embedded into E . We emphasise that property 2 . in the statement of Theorem 4.2 and Corollary 4.3 meansthat the IFS Φ consists of similarities whose contraction ratios are either r or r . Property 3a.means that the similarity ψ j ∗ has the identity matrix as its rotation component. One of thestrengths of Theorem 4.2 and Corollary 4.3 is that they provide information when the elementsof Φ have different contraction ratios. Most results in this area have the additional assumptionthat the elements of Φ have the same contraction ratio (see [1, 2, 8, 12, 13]). Moreover, at thecost of an additional Diophantine condition and rotation assumption, these statements allowsus to weaken the dimension assumption dim H ( E ) < / Proof of Theorem 4.2.
Let Φ and Ψ be two IFSs satisfying the hypothesis of Theorem 4.2.Suppose that F can be affinely embedded into E . Let M be an invertible matrix and a ∈ R d besuch that M ( F ) + a ∈ E. (4.1)We will now set out to prove thatdim H ( E ) ≥ (cid:18) − τ − τ (cid:19) x j ∗ ∈ F denote the unique point satisfying ψ j ∗ ( x j ∗ ) = x j ∗ . Clearly x j ∗ ∈ ψ nj ∗ ( F ) for all n ∈ N . Let y ∗ j be given by y j ∗ := M x j ∗ + a. By (4.1) we know that y ∗ j ∈ E . Therefore there exists a sequence ( i m ) ∈ I N such that y j ∗ =lim m →∞ ϕ i ...i m (0) . Here and throughout we use ϕ i ...i m to denote the concatenation ϕ i ◦· · ·◦ ϕ i m and r i ...i m to denote the product Q ml =1 r i l . Our point y j ∗ satisfies y j ∗ ∈ ϕ i ...i m ( E ) for all m ∈ N .It therefore follows from the above that( M ( ψ nj ∗ ( F )) + a ) ∩ ϕ i ...i m ( E ) = ∅ (4.2)for all n, m ≥
0. Because Φ satisfies the strong separation condition we have c := inf i = i ′ d ( ϕ i ( E ) , ϕ i ′ ( E )) > . It is also the case that for each m ∈ N we have d ( ϕ i ...i m ( E ) , E \ ϕ i ...i m ( E )) ≥ cr i ...i m − . (4.3)It therefore follows from (4.2) and (4.3) that M ( ψ nj ∗ ( F )) + a ⊂ ϕ i ...i m ( E ) whenever Diam ( M ( ψ nj ∗ ( F ))) < cr i ...i m − . (4.4)For m ≥ s m := min (cid:8) n ∈ N : M ( ψ nj ∗ ( F )) + a ⊂ ϕ i ...i m ( E ) (cid:9) . (4.5)It follows from (4.4) that s m < ∞ .We introduce the notation: k M k : = max {| M v | : | v | = 1 }k M k ′ : = min {| M v | : | v | = 1 } . By (4.5) we have k M k ′ ( r ′ j ∗ ) s m Diam ( F ) ≤ Diam ( M ( ψ s m j ∗ ( F ))) ≤ Diam ( ϕ i ...i m ( E )) ≤ Diam ( E ) · r i ...i m . Therefore ( r ′ j ∗ ) s m r i ...i m ≤ Diam ( E ) k M k ′ Diam ( F ) (4.6)for all m ≥
1. Similarly we have( r ′ j ∗ ) s m r i ...i m ≥ c · r ′ j ∗ k M k Diam ( F ) max { r , r } (4.7)when s m ≥
1. Equation (4.7) follows because if it were to fail then we would have
Diam ( M ( ψ s m − j ∗ ( F ))) ≤ k M k ( r ′ j ∗ ) s m − Diam ( F ) < max { r , r } − c · r i ...i m ≤ c · r i ...i m − . Which by (4.4) would imply M ( ψ s m − j ∗ ( F )) + a ⊂ ϕ i ,...,i m ( E ) . This would contradict the defini-tion of s m .It follows from the definition of s m that ϕ − i ...i m ( M ( ψ s m j ∗ ( F )) + a ) ⊂ E. Q m = ( O i ◦ · · · ◦ O i m ) − ◦ M we have r − i ...i m · ( r ′ j ∗ ) s m · Q m ( F ) + a m ⊂ E for some a m ∈ R d . Here we used the fact that the rotation component for ψ j ∗ is the identitymatrix. Therefore r − i ...i m · ( r ′ j ∗ ) s m · Q m ( F − F ) ⊂ E − E (4.8)for m ≥
1. Let v ∈ F − F be a non-zero vector. Such a vector must exists because F isnon-trivial. Then by (4.8) we have r − i ...i m · ( r ′ j ∗ ) s m · Q m v ⊂ E − E (4.9)for all m ≥
1. Using the fact that Q m is the composition of some orthogonal matrices with M ,we see that by taking norms of both sides in (4.9) we have r − i ...i m · ( r ′ j ∗ ) s m · | M v | ∈ {| x − y | : x, y ∈ E } (4.10)for all m ≥
1. Let U := {| x − y | : x, y ∈ E } and V := (cid:8) r − i ...i m ( r ′ j ∗ ) s m | M v | : m ≥ (cid:9) . Consider the map f : (cid:20) c · r ′ j ∗ · | M v |k M k Diam ( F ) max { r , r } , Diam ( E ) · | M v |k M k ′ Diam ( F ) (cid:21) → T given by f ( x ) = log x log r ′ j ∗ mod 1 . The map f is Lipschitz. It now follows from (4.6), (4.7), and the well known fact that Lipschitzmaps cannot increase the upper box dimension (see [7]) thatdim B f ( V ) ≤ dim B ( V ) ≤ dim B ( U ) ≤ dim B ( E − E ) ≤ dim B ( E × E ) = 2 dim H ( E ) . Therefore dim B f ( V )2 ≤ dim H ( E ) . (4.11)Notice that for any m ≥ f (cid:16) r − i ...i m +1 r s m +1 j ∗ | M v | (cid:17) − f (cid:16) r − i ...i m r s m j ∗ | M v | (cid:17) = − log r m +1 log r ′ j ∗ mod 1 . By property 2. the IFS Φ consists of similarities with contraction ratios equal to r or r .Therefore f ( V ) is an αβ orbit for α = − log r log r ∗ j and β = − log r log r ∗ j . Applying Theorem 1.1 and (4.11)we have dim H ( E ) ≥ (cid:18) − τ − τ (cid:19) . This completes our proof. 10 eferences [1] A. Algom,
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