Intersections of multiplicative translates of 3 -adic Cantor sets II: two infinite families
aa r X i v : . [ m a t h . N T ] D ec INTERSECTIONS OF MULTIPLICATIVE TRANSLATES OF -ADIC CANTOR SETS II: TWO INFINITE FAMILIES WILLIAM C. ABRAM, ARTEM BOLSHAKOV, AND JEFFREY C. LAGARIASA
BSTRACT . This paper studies the structure of finite intersections of general multiplica-tive translates C ( M , M , . . . , M n ) = M Σ , ¯2 ∩· · ·∩ M n Σ , ¯2 for integers ≤ M 1. Introduction 21.1. Exceptional set conjecture and nesting constants 41.2. Statistics of ternary digits and n -digit Hausdorff dimension constant 51.3. Roadmap 62. Results 62.1. The infinite family P k = (20 k − Q k = (2 k k − n -digit Hausdorff dimension constants α n . 82.4. Notation 93. Symbolic dynamics, path sets and p -adic path set fractals 93.1. Symbolic dynamics, graphs and finite automata 93.2. p -Adic path sets, sofic shifts and p -adic path set fractals 103.3. p -Adic symbolic dynamics and graph directed constructions 113.4. Interleaving operation on path sets 11 Date : December 5, 2015.The first author received support from an NSF Graduate Research Fellowship. The third author receivedsupport from NSF grants DMS-1101373 and DMS-1401224. 4. The infinite family P k = 2 · k + 1 = (20 k − P k = (20 k − = 2 · k + 1 : Path set structure. 144.2. The Family P k = (20 k − = 2 · k + 1 : Hausdorff dimension. 224.3. Hausdorff dimension bounds for C (1 , P k , ..., P k n ) Q k = 3 k − k + 1 = (2 k k − Q k = (2 k k − = 3 k − k + 1 : Path set structure 245.2. The family Q k = (2 k k − = 3 k − k + 1 : Hausdorff dimension 276. Bounds on Hausdorff dimensions by numbers of ternary digits 276.1. Upper Bound on Γ via n -digit constants α n : Proof of Theorem 2.5. 276.2. Exact bound for α L k = (1 k ) and N k = (10 k − . 309. Appendix B: Relation of families P k = (20 k − and L k +1 = (1 k +1 ) NTRODUCTION Let the -adic Cantor set Σ := Σ , ¯2 be the subset of all -adic integers whose -adic expansions consist of digits and only. This set is a well-known fractal havingHausdorff dimension dim H (Σ ) = log ≈ . . By a multiplicative translate ofsuch a Cantor set we mean a multiplicatively rescaled set r Σ = { rx : x ∈ Σ } , where werestrict to r = pq ∈ Q × being a rational number that is -integral, meaning that r ∈ Z , orequivalently ord ( r ) ≥ . For example the multiplicative translate Σ , ¯1 = 2Σ , ¯2 , whichallows only -adic digits and , has the symbol structure of its digits matching that ofternary expansions of the usual middle-third Cantor set on [0 , .This paper considers sets given as finite intersections of such multiplicative translates: C ( r , r , · · · , r N ) := N \ i =1 r i Σ . (1.1)These sets are fractals and this paper considers the problems of determining their internalstructure and of obtaining bounds on their Hausdorff dimension. The dependence of theHausdorff dimension of the sets C ( r , . . . , r n ) on the parameters ( r , r , . . . , r n ) turns outto be complicated and fascinating.In Part I [3], two of the authors presented a method for exactly computing the Hausdorffdimension of individual sets C ( r , . . . , r n ) . This method is suited for computer experimen-tation. The method is based on the fact all such sets have a special property: the -adicexpansions of members of such a set are characterizable by the set of all infinite paths ina fixed labeled directed graph (finite automaton) that emanate from a fixed initial vertex,where the edge labels are -adic digits. We term sets of this kind, characterized by a finiteautomaton, -adic path set fractals . Two of the authors studied the p -adic version of thisconcept in [2], and showed their Hausdorff dimensions are explicitly computable in termsof properties of the associated finite automaton. p -adic path set fractals in turn are geomet-ric realizations of objects in symbolic dynamics called path sets . Forgetting the geometricdata associated to a p -adic path set fractal Y , that is, thinking of the -adic digits as analphabet with no additional structure, recovers an underlying path set X which is the set ofall infinite strings of digits from { , , . . . , p − } corresponding to elements of Y . The pathset underlying the -adic path set fractal C ( r , . . . , r n ) is denoted X ( r , . . . , r n ) , and will NTERSECTIONS OF MULTIPLICATIVE TRANSLATES OF -ADIC CANTOR SETS II: TWO INFINITE FAMILIES3 play a role in the results of this paper. The papers [2], [3] gave between them algorithmsto effectively compute X ( r , . . . , r n ) when given ( r , r , ..., r n ) . Section 3 reviews basicresults on path sets and p -adic path set fractals; a general theory of path sets was previouslydeveloped by two of the authors in [1].This paper is concerned with the case C (1 , M ) for M a positive integer. The Hausdorffdimension dim H ( C (1 , M )) has a clear dependence on certain simple properties of theternary expansion ( M ) of M . For example Part I observed:(i) dim H ( C (1 , M )) = 0 whenever the last ternary digit of ( M ) is a , i.e. M ≡ .(ii) dim H ( C (1 , M )) = dim H ( C (1 , M )) . In consequence, all trailing zeros in thebase expansion of M may be cancelled off without changing the Hausdorff di-mension.However the dependence on M seems anything but simple when examined more closely.It appears that arithmetic properties of M influence both the structure of the underlyingautomata and the Hausdorff dimension in extremely complex ways. Part I treated in detailtwo infinite families of M whose ternary expansion ( M ) had a particularly simple form,where an exact answer for the Hausdorff dimension could be obtained.(1) M = L k = (1 k ) , that is L k = (3 k − . It obtained a Hausdorff dimensionformula for each k ≥ and deduced that dim H ( C (1 , L k )) → as n → ∞ ([3,Theorem 5.2]).(2) M = N k = (10 k − , that is N k = 3 k + 1 . It showed for each k ≥ that dim H ( C (1 , N k )) = log φ ≈ . , where φ = √ ([3, Theorem 5.5]).The automata associated to the second of these families displayed considerable complexity.The automaton associated to N k had a number of states growing exponentially with k andwas strongly connected; it is remarkable that its Perron eigenvalue could be computedexactly. Salient facts on these families are collected in Appendix A (Section 8) for easyreference.This paper continues the study of the sets C (1 , M ) for various integers M ≥ . Weobtain results for two new infinite families of M having ternary expansions ( M ) of a reg-ular form, P k = 2 · k + 1 = (20 k − and Q k = 3 k − k + 1 = (2 k k − ; they arestated in Section 2. When compared to the families treated in Part I, these families revealadditional complexity in the structure of the associated automata and the behavior of theHausdorff dimension. In particular the automata associated to one of these families arenot strongly connected; they are reducible and have arbitrarily large numbers of stronglyconnected components. We bound the Hausdorff dimension of such C (1 , M ) through esti-mation of the Perron eigenvalue of the adjacency matrix of these automata. To estimate theHausdorff dimension of one family, we make use of an operation on path sets termed inter-leaving , that we introduce in Section 3.4. The structure of the automata was first guessedfrom computer experiments and then proved. In addition to studying these two families thepaper presents further results from computer experiments to test the relation of Hausdorffdimension to particular patterns in the ternary expansion of M .The original motivation for studying questions of this kind arose from a problem ofErd˝os [8]. This problem was generalized to a question over the -adic integers by thethird author ([12]), who proposed a weaker version of the Erd˝os problem, the Exceptionalset conjecture , explained below, which asserts that a certain set has Hausdorff dimension . The results of this paper yield new information about the Exceptional set conjecturewithout resolving it, see Section 1.2. WILLIAM C. ABRAM, ARTEM BOLSHAKOV, AND JEFFREY C. LAGARIAS Exceptional set conjecture and nesting constants. Erd˝os [8] conjectured that forevery n ≥ , the ternary expansion of n does not omit the digit . A weak version of thisconjecture asserts that there are only finitely many n such that the ternary expansion of n does not omit the digit . Both versions of this conjecture are open and appear difficult.In [12] the third author proposed a -adic generalization of this problem, as follows. Let Z denote the -adic integers, and let a -adic integer α have -adic expansion ( α ) := a + a · a · + · · · , with all a i ∈ { , , } . It introduced the following notion. Definition 1.1. The -adic exceptional set E ( Z ) is given by E ( Z ) := { λ ∈ Z : for infinitely many n ≥ the expansion (2 n λ ) omits the digit } . This definition is less stringent than the Erd˝os problem in allowing variation of the newparameter λ . The weak version of Erd˝os’s conjecture above is equivalent to the assertionthat / ∈ E ( Z ) .That paper proposed the following conjecture [12, Conjecture 1.7]. Conjecture 1.2. (Exceptional Set Conjecture) The -adic exceptional set E ( Z ) has Haus-dorff dimension zero, i.e. dim H ( E ( Z )) = 0 . (1.2)Clearly ∈ E ( Z ) , and our state of ignorance is such that we do not know whether E ( Z ) = { } or not. In [12] the Exceptional Set Conjecture was approached by introduc-ing the sets E ( k ) ( Z ) := { λ ∈ Z : at least k values of (2 n λ ) omit the digit 2 } , (1.3)which yield the containment relation E ( Z ) ⊆ ∞ \ k =1 E ( k ) ( Z ) . (1.4)That paper obtained the upper bound dim H ( E ( Z )) ≤ dim H ( E (2) ( Z )) ≤ . The sets E ( k ) ( Z ) form a nested family Σ , ¯2 = E (1) ( Z ) ⊇ E (2) ( Z ) ⊇ E (3) ( Z ) ⊇ · · · , and are themselves expressed in terms of intersection sets (1.1) as E ( k ) ( Z ) = [ ≤ m <... The (dyadic) nesting constant Γ is given by Γ := lim k →∞ dim H ( E ( k ) ( Z )) . (1.6)The containment relation (1.4) implies that the nesting constant upper bounds to theHausdorff dimension of the exceptional set, dim H ( E ( Z )) ≤ Γ . (1.7) NTERSECTIONS OF MULTIPLICATIVE TRANSLATES OF -ADIC CANTOR SETS II: TWO INFINITE FAMILIES5 The third author raised the question in [12] whether Γ = 0 , which if true would imply theExceptional Set Conjecture. This question is currently unanswered.Part I [3, Section 1.2] approached the problem of obtaining improved upper bounds for Γ by introducing a relaxed upper bound Γ ⋆ , called there the generalized nesting constant ,obtained by replacing C (2 m , . . . , m k ) with C (1 , M , ..., M k − ) in the definition above.That paper showed Γ ≤ Γ ⋆ ≤ , and also established the lower bound Γ ⋆ ≥ 12 log φ ≈ . . It follows that one cannot resolve whether Γ = 0 or not using the relaxation Γ ⋆ . Statistics of ternary digits and n -digit Hausdorff dimension constant. A focusof this work was to shed light on the Exceptional set conjecture, by gathering evidencewhether there might exist simple statistics of the ternary expansion ( M ) of a single integer M which will predict that the Hausdorff dimension dim H ( C (1 , M )) must go to as thevalue of the statistic goes to infinity.In this paper we resolve this question for the statistic d ( M ) that counts the number ofnonzero digits in the ternary expansion of the positive integer ( M ) . This value coincideswith the number of nonzero digits in the -adic expansion of M ; note that a -adic integer α has a finite number of non-zero digits if and only if it is a non-negative integer α ∈ N . Definition 1.4. The n -digit Hausdorff dimension constant α n is given by α n := sup M ≥ { dim H ( C (1 , M )) : The expansion ( M ) has at least n nonzero ternary digits } . By definition the α n form a nonincreasing sequence of nonnegative numbers, so thatthe limit Γ ⋆⋆ := lim n →∞ α n exists. Known results in number theory, detailed in Section 6, imply that the number ofnonzero ternary digits of n diverges as n goes to infinity. Thus, we obtain an upper boundon the dyadic nesting constant Γ ≤ Γ ⋆⋆ = lim n →∞ α n = inf n α n . (1.8)One of the infinite families studied in this paper has d ( M k ) → ∞ as k → ∞ and using itwe show Γ ⋆⋆ = inf n α n = log √ ! ≈ . . (1.9)In particular by (1.7) we obtain an improved upper bound for the Hausdorff dimension ofthe exceptional set dim H ( E ( Z )) ≤ Γ ≤ Γ ⋆⋆ ≤ log √ ! ≈ . . (1.10)In the opposite direction (1.9) establishes that the statistic d ( M ) does not have the prop-erty that the Hausdorff dimension must go to as the statistic d ( M ) → ∞ .The final section of the paper empirically studies the Hausdorff dimension of C (1 , M ) with respect to two other simple statistics of the ternary expansion ( M ) : the block number b ( M ) and intermittency s ( M ) ; these satisfy b ( M ) ≤ s ( M ) . These are defined inSection 7. WILLIAM C. ABRAM, ARTEM BOLSHAKOV, AND JEFFREY C. LAGARIAS Roadmap. Section 2 states the main results. Section 3 reviews properties of p -adicpath sets and their symbolic dynamics, drawing on [1] and [2]. Intersections of multi-plicative translates of -adic Cantor sets are a special case of these constructions. Section3.4 introduces an interleaving operation on path sets and analyzes its effect on Hausdorffdimension. Section 4 studies the sets C (1 , P k ) for the infinite family P k , analyzes thestructure of their associated automata, and proves Theorems 2.1-2.2, and additional re-sults. Section 5 studies the structure of C (1 , Q k ) for the infinite family Q k , and provesTheorems 2.3-2.4. Section 6 deals with results on the quantities α n and proves Theo-rems 2.5-2.6. Section 7 presents empirical results on Hausdorff dimensions of C (1 , M ) for M having specified statistics of their ternary expansions ( M ) .Appendix A (Section 8) describes results for two infinite families C (1 , L k ) and C (1 , N k ) treated in Part I [3]. Appendix B (Section 9) relates Hausdorff dimensions of C (1 , P k ) tothose of C (1 , L k +1 ) . Acknowledgments. We thank Yusheng Luo for an important observation on the structureof the automata for the sets P k , incorporated in Definition 4.3 and Proposition 4.4. W.A. thanks the University of Michigan, where much of this work was carried out. W. A.and A. B. would also like to thank Ridgeview Classical Schools, which facilitated theircollaboration. W.A. was partially supported by an NSF graduate fellowship. J. L. wassupported by NSF grants DMS-1101373 and DMS-1401224. Some work of J.L. on thepaper was done at ICERM, where he received support from the Clay Foundation as a ClaySenior Scholar. He thanks ICERM for support and good working conditions.2. R ESULTS The main results of this paper consist of determination of presentations of the -adicpath sets X (1 , P k ) and X (1 , Q k ) associated to members of two infinite families C (1 , P k ) and C (1 , Q k ) given below, with estimates of their Hausdorff dimensions, along with exper-imental results for dim H ( C (1 , M )) for certain other M presented in Section 7.2.1. The infinite family P k = (20 k − . We study the path set structure of families ofintegers having few nonzero ternary digits. The only infinite families of numbers havingexactly two nonzero ternary digits and dim H ( C (1 , N )) > are N k = 3 k + 1 = (10 k − and P k = (20 k − = 2 · k + 1 . The family N k was studied in Part I and here we studythe family P k .We directly compute the Hausdorff dimensions of the first few sets C (1 , P k ) using thealgorithms of Part I to be the following. NTERSECTIONS OF MULTIPLICATIVE TRANSLATES OF -ADIC CANTOR SETS II: TWO INFINITE FAMILIES7 Path Set P k Vertices Perron eigenvalue Hausdorff dim C (1 , P ) . . C (1 , P ) 19 8 . . C (1 , P ) 55 16 . . C (1 , P ) 163 32 . . C (1 , P ) 487 64 . . C (1 , P ) . . C (1 , P ) . . C (1 , P ) . . TABLE 2.1. Hausdorff dimension of C (1 , P k ) (to six decimal places)The first thing to observe from this data is the non-monotonic behavior of the Hausdorffdimension as a function of k ; the second observation is the possibility that the dimensionsare bounded away from zero. Our results below explain both these features. We alsoobserve that dim H ( C (1 , P k )) = dim H ( C (1 , L k +1 )) for ≤ k ≤ but equality doesnot hold for k = 5 . In an Appendix B (Section 9) we show that dim H ( C (1 , P k )) ≥ dim H ( C (1 , L k +1 )) holds in general.Our first result determines properties of a presentation of the path set X (1 , P k ) . Theresulting directed graphs are shown to be reducible, having a complicated structure withnested strongly connected components. Theorem 2.1. (Path set presentation for family P k ) (1) For P k = 2 · k + 1 = (20 k − , the path set X (1 , P k ) underlying C (1 , P k ) has apath set presentation ( G k , v ) that has exactly k +1 vertices.(2) The graph G k is a nested sequence of ⌊ k/ ⌋ distinct strongly connected compo-nents.(3) The underlying graph G = G k for G k has an automorphism of order and is aconnected double cover of its quotient graph H k . The structure of G k is that of a “Matryoshka doll" with a single set of nested componentsat each level. The non-monotonicity of the Hausdorff dimension as a function of k can berelated to the existence of multiple strongly connected components in the graphs G k . Thenon-monotonicity occurs because of a switch in which strongly connected component hasthe largest topological entropy. We discuss this issue further in Section 4.2, see Remark4.6.Regarding the behavior of the Hausdorff dimension as k → ∞ , we establish the follow-ing result. Theorem 2.2. (Hausdorff dimension bounds for family P k = 2 · k + 1 ) (1) The Hausdorff dimension of C (1 , P k ) satisfies the asymptotic lower bound lim inf k →∞ dim H ( C (1 , P k )) ≥ 18 log (2) . (2) Furthermore, for all k ≥ , dim H ( C (1 , P k )) ≥ 113 log (2) . The lower bounds in Theorem 2.2 are obtained by further inspection of the graph asso-ciated to C (1 , P k ) . We also have an upper bound dim H ( C (1 , P k )) ≤ log φ. WILLIAM C. ABRAM, ARTEM BOLSHAKOV, AND JEFFREY C. LAGARIAS which follows from Theorem 6.2 below.In Section 4.3 we obtain additional results on intersection of sets in the infinite family P k above. We show that the Hausdorff dimensions of arbitrarily large intersections arealways positive. However this is no longer true if we allow intersections of sets from theinfinite family P k with those of the infinite family N k = (10 k − treated in [3, Sect. 4]and reviewed in Appendix A (Section 8), which also consists of numbers having exactlytwo nonzero ternary digits. For example, it is easy to show that for each k ≥ , C (1 , N k , P k ) = { } , so that dim H ( C (1 , N k , P k )) = 0 .2.2. The infinite family Q k = (2 k k − . We next study an infinite family of integerswhose number of nonzero ternary digits grows without bound: Q k = (2 k k − =3 k − k + 1 . The example Q having a large Hausdorff dimension was found by computersearch, and led to study of this family. Theorem 2.3. (Path set presentation for family Q k )(1) For Q k = 3 k − k + 1 = (2 k k − , the path set X (1 , Q k ) underlying C (1 , Q k ) has a path set presentation ( G k , v ) that has exactly k vertices and · k − edges.(2) The underlying graph G k is strongly connected. Though the number of nonzero ternary digits of Q k grows without bound, the Hausdorffdimension of C (1 , Q k ) is constant independent of k . Theorem 2.4. (Hausdorff dimensions for family Q k = 3 k − k + 1 ) For all k ≥ theHausdorff dimension of C (1 , Q k ) satisfies dim H ( C (1 , Q k )) = log φ ≈ . , where φ = √ . This result is established by showing that the path set X (1 , Q k ) is given by an interleav-ing construction from the path set X (1 , Q ) , that is X (1 , Q k ) = X (1 , ( ∗ k ) , as defined inSection 3.4.2.3. The n -digit Hausdorff dimension constants α n . It is a known fact that the numberof nonzero ternary digits in (2 n ) goes to infinity as n → ∞ , i.e. for each k ≥ there areonly finitely many n with (2 n ) having at most k nonzero ternary digits. Using this factwe easily deduce the following consequence. Theorem 2.5. The nesting constant Γ satisfies Γ ≤ lim n →∞ α n . (2.1) In particular dim H ( E ( Z )) ≤ Γ ∗∗ = lim n →∞ α n . It follows that individual values α n give upper bounds on Γ . Theorem 2.6. We have for all k ≥ that α k = log φ ≈ . , where φ = √ is the golden ratio. This value is attained by C (1 , Q k ) for Q k := (2 k k − . NTERSECTIONS OF MULTIPLICATIVE TRANSLATES OF -ADIC CANTOR SETS II: TWO INFINITE FAMILIES9 In particular this result yields an improved upper bound on the nesting constant Γ ≤ log φ, and on the Hausdorff dimension of the Exceptional set. It also gives Γ ⋆⋆ = log φ ≈ . . We prove Theorem 2.6 in Section 6.2.Using the known bound for the generalized dyadic nesting constant Γ ⋆ ≤ α establishedin Part I [3, (1.16)] we obtain the following corollary. Corollary 2.7. We have Γ ∗ ≤ log φ ≈ . , in which φ = √ is the golden ratio. Notation. The notation ( m ) means either the base expansion of the positive inte-ger m , or else the -adic expansion of ( m ) . In the -adic case this expansion is to be readright to left, so that it is compatible with the ternary expansion. That is, α = P ∞ j =0 a j j will be written ( · · · a a a ) .3. S YMBOLIC DYNAMICS , PATH SETS AND p - ADIC PATH SET FRACTALS Symbolic dynamics, graphs and finite automata. The constructions of this paperare based on the fact that the points in intersections of multiplicative translates of -adicCantor sets have -adic expansions that are describable in terms of allowable paths gener-ated by finite directed labeled graphs. We use symbolic dynamics on certain closed subsetsof the one-sided shift space Σ = A N with fixed symbol alphabet A , which for our ap-plication will be specialized to A = { , , } . A basic reference for directed graphs andsymbolic dynamics, which we follow, is Lind and Marcus [14].By a graph we mean a finite directed graph, allowing loops and multiple edges. A labeled graph is a graph assigning labels to each directed edge; these labels are drawn froma finite symbol alphabet. A labeled directed graph can be interpreted as a finite automaton in the sense of automata theory. In our applications to -adic digit sets, the labels are drawnfrom the alphabet A = { , , } . In a directed graph, a vertex is a source if all directededges touching that vertex are outgoing; it is a sink if all directed edges touching that edgeare incoming. A vertex is essential if it is neither a source nor a sink; and is called stranded otherwise. A graph is essential if all of its vertices are essential. A graph G is stronglyconnected if for each two vertices i, j there is a directed path from i to j . We let SC ( G ) denote the set of strongly connected component subgraphs of G .We use some basic facts from the Perron-Frobenius theory of nonnegative matrices.The Perron eigenvalue ([14, Definition 4.4.2]) of a nonnegative real matrix A = 0 is thelargest real eigenvalue β ≥ of A . A nonnegative matrix is irreducible if for each rowand column ( i, j ) some power A m has ( i, j ) -th entry nonzero. A nonnegative matrix A is primitive if some power A k for an integer k ≥ has all entries positive; primitivity impliesirreducibility but not vice versa. The Perron-Frobenius Theorem [14, Theorem 4.2.3] foran irreducible nonnegative matrix A states that:(1) The Perron eigenvalue β is geometrically and algebraically simple, and has aneverywhere positive eigenvector v . (2) All other eigenvalues µ have | µ | ≤ β , so that β = σ ( A ) , the spectral radius of A .(3) Any other everywhere positive eigenvector must be a positive multiple of v . For a general nonnegative real matrix A = 0 , the Perron eigenvalue need not be simple,but it still equals the spectral radius σ ( A ) and it has at least one everywhere nonnegativeeigenvector.We apply this theory to adjacency matrices of graphs. A (vertex-vertex) adjacencymatrix A = A G of the directed graph G has entry a ij counting the number of directededges from vertex i to vertex j . The adjacency matrix is irreducible if and only if theassociated graph is strongly connected, and we also call the graph irreducible in this case.Here primitivity of the adjacency matrix of a directed graph G is equivalent to the graphbeing strongly connected and aperiodic, i. e. the greatest common divisor of its (directed)cycle lengths is . For an adjacency matrix of a graph containing at least one directedcycle, its Perron eigenvalue is necessarily a real algebraic integer β ≥ (see Lind [13] fora characterization of these numbers).3.2. p -Adic path sets, sofic shifts and p -adic path set fractals. Our basic objects arespecial cases of the following definition. A pointed graph is a pair ( G , v ) consisting of adirected labeled graph G = ( G, E ) and a marked vertex v of G . Here G is a (directed)graph and E is an assignment of labels ( e, ℓ ) = ( v , v , ℓ ) to the edges of G , where everyedge gets a single label, and no two triples are the same (but multiple edges and loops arepermitted otherwise). Definition 3.1. Given a pointed graph ( G , v ) its associated path set P = X G ( v ) ⊂ A N is the set of all infinite one-sided symbol sequences ( x , x , x , ... ) ∈ A N , giving thesuccessive labels of all one-sided infinite walks in G issuing from the distinguished vertex v . Many different ( G , v ) may give the same path set P , and we call any such ( G , v ) a presentation of P .An important class of presentations have the following extra property. We say that a di-rected labeled graph G = ( G, v ) is right-resolving if for each vertex of G all directed edgesoutward have distinct labels. (In automata theory G is called a deterministic automaton .)One can show that every path set has a right-resolving presentation.Note that the labeled graph G without a marked vertex determines a one-sided sofic shift in the sense of symbolic dynamics, as defined in [1]. This sofic shift comprises the setunion of the path sets at all vertices of G . Path sets are closed sets in the shift topology, butare in general non-invariant under the one-sided shift operator. Those path sets P that areinvariant are exactly the one-sided sofic shifts [1, Theorem 1.4].We study the path set concept in symbolic dynamics in [1]. The collection of path sets P = X G ( v ) in a given alphabet is closed under finite union and intersection ([1, Theorem1.2]). The symbolic dynamics analogue of Hausdorff dimension is topological entropy.The topological entropy of a path set H top ( P ) is given by H top ( P ) := lim sup n →∞ n log N n ( P ) , where N n ( P ) counts the number of distinct blocks of symbols of lengh n appearing inelements of P . The topological entropy is easy to compute given a right-resolving presen-tation. By [1, Theorem 1.13], it is H top ( P ) = log β (3.1)where β is the Perron eigenvalue of the adjacency matrix A = A G of the underlyingdirected graph G of G , e.g. the spectral radius of A . NTERSECTIONS OF MULTIPLICATIVE TRANSLATES OF -ADIC CANTOR SETS II: TWO INFINITE FAMILIES11 p -Adic symbolic dynamics and graph directed constructions. We now suppose A = { , , , ..., p − } . We can view the elements of a path set P on this alphabetgeometrically as describing the digits in the -adic expansion of a -adic integer. This isdone using a map φ : A N → Z p from symbol sequences into Z p . We call the resultingimage set K = φ ( P ) a p -adic path set fractal . Such sets are studied in [2], where theyare related to graph-directed fractal constructions. The class of p -adic path set fractalsis closed under the Minkowski sum and p -adic addition and multiplication by rationalnumbers r ∈ Q that lie in Z p ([2, Theorems 1.2-1.4]).It is possible to compute the Hausdorff dimension of a p -adic path set fractal directlyfrom a suitable presentation of the underlying path set P = X G ( v ) . We will use thefollowing result. Proposition 3.2. Let p be a prime, and K a set of p -adic integers whose allowable p -adicexpansions are described by the symbolic dynamics of a p -adic path set X K on symbols A = { , , , · · · , p − } . Let ( G , v ) be a presentation of this path set that is right-resolving.(1) The map φ p : Z p → [0 , taking α = P ∞ k =0 a k p k ∈ Z p to the real number withbase p expansion φ p ( α ) := P ∞ k =0 a k p k +1 is a continuous map, and the image of K under thismap, K ′ := φ p ( K ) ⊂ [0 , , is a graph-directed fractal in the sense of Mauldin-Williams.(2) The Hausdorff dimension of the p -adic path set fractal K is dim H ( K ) = dim H ( K ′ ) = log p β, (3.2) where β is the spectral radius of the adjacency matrix A of G .Proof. These results are proved in [2, Section 2]. (cid:3) In this paper we treat the case p = 3 with A = { , , } . The -adic Cantor set is a -adic path set fractal, so these general properties above guarantee that the intersection ofa finite number of multiplicative translates of -adic Cantor sets will itself be a -adic pathset fractal K , generated from an underlying path set.To do calculations with such sets we will need algorithms for converting presentationsof a given p -adic path set to presentations of new p -adic path sets derived by the operationsabove. We refer the reader to [2] for the p -adic arithmetic operations, and to [1] for unionand intersection. A further useful operation called interleaving will be developed in thenext subsection; this operation is sometimes useful in computing Hausdorff dimension.3.4. Interleaving operation on path sets. Let P = X G ( v ) ⊂ A N be a path set, and let n be a positive integer. In the paper [1] the first and third authors studied a decimation operation on path sets. Given j ≥ and m ≥ , define the decimation map ψ j,m : A N →A N by ψ j,m ( a a a · · · ) := ( a j a j + m a j +2 m · · · ) . The decimation operation extracts the digits of the path set in a specified infinite arithmeticprogression of indices. We set ψ j,m ( P ) := { ψ j,m ( x ) : x ∈ P} . Here [1, Theorem 1.5] proved that if P is a path set, then for each fixed ( j, m ) with j ≥ , m ≥ the sets ψ j,m ( P ) are path sets.Here we consider a kind of inverse operator to decimation, which we term interleaving . Definition 3.3. Let n ≥ be given. The n -interleaving of a closed set X ⊂ A N (notnecessarily a path set) is X ( ∗ n ) := { ( x i ) ∞ i =0 ∈ A N : ( x j , x j + n , x j +2 n , · · · ) ∈ X for all ≤ j ≤ n − } . We will show that the interleaving P ( ∗ n ) is itself a path set, and that its topologicalentropy is the same as that of P . Proposition 3.4. (1) For any n ≥ and any path set P , the n -interleaving set P ( ∗ n ) is apath set.(2) There is an algorithm taking n and a path set presentation G of P and giving a pathset presentation H of P ( ∗ n ) . If G has k verticies and m edges, then H has k n verticies and mk n − edges.Proof. It suffices to prove (2). Suppose P = X G ( v ) , and that the vertices of G are v , v , . . . , v k − , so that G has k vertices. Let l j be the label of vertex v j for each ≤ j ≤ k − . If the l j do not all have the same number of digits, append ′ s to the left of labelsas necessary to ensure that the labels l , . . . , l j are distinct and have the same number ofdigits.The vertex set of H will be V = { v i ,i ,...,i n | ≤ i j ≤ k − for all j } , so that H willhave k n vertices. The vertex v i ,i ,...,i n will have label l = l i ⋆ l i ⋆ · · · ⋆ l i n , that is, theconcatenation of the labels of v i , v i , . . . , v i n . Since the labels l j are all distinct and havethe same number of digits, the vertex labels in H as defined will also be distinct.Now for each edge labeled a from v i to v j in G , construct an edge labeled a from v i ,i ,...,i n − ,i to v j,i ,i ,...,i n − for all ≤ i , i , . . . , i n − ≤ k − . Thus, for eachedge of G , H will have k n − corresponding edges, so that if G has m edges, then H has mk n − edges. H is evidently right-resolving or strongly connected if G is right-resolvingor strongly connected, respectively. For simplicity, we will assume from here that G isright-resolving. We can do this since if G is not right-resolving, we can perform the right-resolving construction of [1, Section 3] to obtain a right-resolving presentation of P , andproceed with this presentation in place of G .We claim that P ( ∗ n ) = X H ( v , ,..., ) . First we will show that P n ⊆ X H ( v , ,..., ) .Suppose ( x t ) ∞ t =0 ∈ P n . Then there must be elements ( x ,t ) ∞ t =0 , ( x ,t ) ∞ t =0 , . . . , ( x n − ,t ) ∞ t =0 ∈ P such that x j,t = x nt + j for all ≤ j ≤ n − and ≤ t < ∞ . Since G is right-resolving,each of these elements of P corresponds to a unique infinite vertex path v , v i j, , v i j, , . . . in G . We can traverse an initial path in the pointed graph H ( v , , ,..., ) with labels x , x , . . . , x n − , since there are edges with each of these labels emanating from v in G .This path takes us to the vertex v i n − , ,i n − , ,...,i , . Since there is a vertex labeled x n + j emenating fom vertex v i j, and going to v i j, for all ≤ j ≤ n − , we can extend our pathto a path labeled x , x , . . . , x n − beginning at v , ,..., and ending at v i n − , ,i n − , ,...,i , .Inductively, assume we have constructed a path with labels x , x , . . . , x rn − in H originating at v , ,..., and terminating at v i n − ,r − ,i n − ,r − ,...,i ,r − . Then since there isan edge in G labeled x rn + j from v j,r − to v j,r , we can extend our path to a path labeled x , x , . . . , x ( r +1) n − terminating at v i n − ,r ,i n − ,r ,...,i ,r . Thus, there is an infinite path in H originating at v , ,..., with label ( x , x , x , . . . ) , so ( x i ) ∞ i =0 ∈ X H ( v , ,..., ) , hence P n ⊆ X H ( v , ,..., ) .Now to show X H ( v , ,..., ) ⊆ P n : Suppose ( x i ) ∞ i =0 is an element of X H ( v , ,..., ) .Then there is a vertex path v , ,..., ; v i , ,..., ; v i ,i , ,..., ; . . . ; v i n − ,i n − ; ...,i ; . . . in H which can be traversed by edges labeled x , x , . . . . Notice that the first coordinate ofa vertex must be the last coordinate of the vertex that follows after n − steps. Since theinitial vertex is v , ,..., , we know that for each ≤ j ≤ n − , there is an edge in G labeled x j from v to v i j . For any j < ∞ , an edge in H labeled x j from v i ,i ,...,i n to v i n +1 ,i ,i ,...,i n − corresponds to an edge in G labeled x j fom v i n to v i n +1 . Following our NTERSECTIONS OF MULTIPLICATIVE TRANSLATES OF -ADIC CANTOR SETS II: TWO INFINITE FAMILIES13 path in H for n − more steps gets us to a vertex whose last coordinate is i n +1 , so the edgein H labeled x n + j emanating from this vertex corresponds to an edge in G labeled x n + j emanating from v i n +1 . Thus, for each ≤ j ≤ n − , the labels ( x j , x j + n x j +2 n , . . . ) arethe labels of an infinite path in G originating at v , so ( x i ) ∞ i =0 ∈ P n , hence X H ( v , ,..., ) ⊆P n , as desired. (cid:3) Remark . (1) The presentation H of P ( ∗ n ) given in the proof above is right-resolving(resp. strongly connected) if and only if the presentation G of P used in its construction isright-resolving (resp. strongly connected).(2) The operation of interleaving can be extended to interleave several different sets I ( X , X , ..., X m ) := { x ∈ A N : ψ j,m ( x ) ∈ X i for ≤ j ≤ m − . } One can show that if each X i = P i is a path set then I ( P , P , · · · , P n ) is a path set.We next show that the n -interleaving operation P ( ∗ n ) has the nice feature that it pre-serves topological entropy. Following [1] we define the path topological entropy H p ( P ) of a path set P by H p ( P ) := lim sup k →∞ k log N Ik ( P ) , (3.3)where N Ik ( P ) is the number of initial blocks of length k from P , then [1, Theorem 1.11]shows that H p ( P ) = H top ( P ) , (3.4)and that the lim sup ’s are obtained as limits. Proposition 3.6. If P is a path set, then H top ( P ( ∗ n ) ) = H top ( P ) . (3.5) Proof. Using (3.4), it suffices to show that P and P ( ∗ n ) have the same path entropy. Butwe can see directly from the definition of P ( ∗ n ) that N Ink ( P ( ∗ n ) ) = ( N Ik ( P )) n , since aninitial path of length nk in P ( ∗ n ) corresponds to n (not necessarily distinct) initial paths oflength k in P . Thus, H p ( P ( ∗ n ) ) = lim k →∞ k log N Ik ( P ( ∗ n ) )= lim k →∞ nk log N Ink ( P ( ∗ n ) )= lim k →∞ nk log[( N Ik ( P )) n ]= lim k →∞ k log N Ik ( P ) = H p ( P ) , as desired. (cid:3) If A = { , , . . . , p − } , let φ : A N → Z p be the map of Section 3.3, which maps thepath set P to the corresponding p -adic path set fractal K = φ ( P ) . We have the followingCorollary. Corollary 3.7. If P is a path set on the alphabet A = { , , , . . . , p − } , then the p -adicpath set fractals K = φ ( P ) and K ′ = φ ( P ( ∗ n ) ) have the same Hausdorff dimension.Proof. This follows immediately from (3.1), Proposition 3.6, and Proposition 3.2. (cid:3) Remark . (1) Corollary 3.7 is useful in computing Hausdorff dimensions of path setsin our examples. Let P = X (1 , be the Golden Mean Shift, which is also the path setunderlying the -adic path set fractal C (1 , . An element of C (1 , N k ) = C (1 , (10 k − ) is any -adic integer consisting of ’s and ’s and for which no is followed k digits laterby another . Recognizing this property allows us to see for N k = (10 k − = 3 k +1 thatthe path set X (1 , N k ) underlying C (1 , N k ) is just P ( ∗ k ) . Corollary 3.7 provides anotherproof of a result in part I ([3, Theorem 5.5]) asserting that dim H ( C (1 , N k )) = log φ , sincethis now follows from the basic computation dim H ( C (1 , φ . One may comparethis argument to the proof given in [3, Theorem 5.5]. Let G be the presentation of C (1 , given by Algorithm A of [3]. The algorithm of Proposition 3.4 applied to k and G andAlgorithm A of [3] give isomorphic graph presentations of C (1 , N k ) .(2) In Section 5 below, we will prove Theorem 2.4, which states that dim H ( C (1 , Q k )) = log φ, by a similar argument.4. T HE INFINITE FAMILY P k = 2 · k + 1 = (20 k − We obtain a relatively complete description of the path set structure for the family P k =2 · k + 1 = (20 k − . As a preliminary we review results for the infinite families L k and N k studied in part I ([3, Section 4]).4.1. The Family P k = (20 k − = 2 · k +1 : Path set structure. We study the structureof a path set presentation of the -adic expansions of elements in C (1 , P k ) . The followingexample gives a path set presentation for P = 19 . Example 4.1. A path set presentation of the path set X (1 , associated to C (1 , , with 19 = (201) , is shown in Figure 4.1. The vertex labeled is the marked initial vertex. NTERSECTIONS OF MULTIPLICATIVE TRANSLATES OF -ADIC CANTOR SETS II: TWO INFINITE FAMILIES15 01 1011 1 00 01 0FIGURE 4.1. Path set presentation of X (1 , . The marked vertex is .The graph in Figure 4.1 has adjacency matrix A = , which has Perron eigenvalue β ≈ . , so dim H ( C (1 , β ≈ . . An important feature of the graph in Figure 4.1 is that it is reducible with two stronglyconnected components, one component being the nodes in the middle, and the other thering of nodes around the outside. The (oriented) dependency graph of the strongly con-nected components is a tree with nodes. The Perron eigenvalue β of the graph above is associated with the outer strongly connected component with nodes. The inner compo-nent has topological entropy .We describe the path set presentation in general. The vertex labels of the presentationwill be described using the following definition. Definition 4.1. Classify the labels of the vertices in the graph G k as numbers m with ≤ m ≤ k whose finite -adic expansions (read right to left) are of types (S1) and (S2)given by:(S1) The expansion ( X ) , written with exactly k digits, omits the digit .(S2) The -adic expansion of m contains a single digit , and has the form ( X j ) for some ≤ j ≤ k , with ( X j ) written with exactly k digits, plus m = 3 k =(10 k ) .Note that an (S2) label has initial -adic digits consisting of a string of zeros, followedby a . Proposition 4.2. For P k = 2 · k + 1 the path set X (1 , P k ) associated to C (1 , P k ) has apresentation ( G k , v ) with the following properties.(1) The vertices v m have labels m consisting of those ≤ m ≤ k whose -adicexpansion ( m ) is one of the two types (S1) and (S2) above.(2) The underlying directed graph G of G k has exactly k +1 vertices.(3) The reflection map R ( m ) = 3 k − m which acts on vertex labels of the underlyingdirected graph G k is an automorphism of G k . Given any path from (0) to vertex m , thereis a directed path from vertex (10 k ) to vertex k − m of the same length, visiting the set ofreflected vertices of the original path, and having all the edge labels reversed (exchanging and ).Proof. The presentation found in this theorem will be that given by the construction ofAlgorithm A in part I [3].From the proof of Theorem 9.1 we know that a vertex with label m = 3 k is reachableby a directed path from vertex m = 0 and vice-versa.We prove the proposition by showing, in order:(G1) The vertices of G reachable from v have labels ≤ m ≤ k which are a subsetof the labels (S1) and (S2).(G2) The set of vertex labels m satisfying (S1) or (S2) are exchanged under the reflec-tion map R ( m ) = 3 k − m . The set of all possible m satisfying (S1), respectively(S2), each have cardinality k .(G3) Each path emanating from vertex m = 0 corresponds to a unique path emanatingfrom vertex m = 3 k with the new path having reflected vertex labels and reversededge labels, and vice versa.(G4) The set of all reachable vertices is invariant under the reflection map.(G5) All vertices with labels of type (S1) are reachable.(G6) The reflection map on vertices induces a graph automorphism of G of order withno fixed points. Thus G is a double cover of the resulting quotient graph H .To establish (G1) we proceed by induction on the length n of a shortest path to a givenvertex. The base case m = 0 is an (S1) label. Following a single edge changes a vertexlabel ( Xs ) (with s = 0 , , ) to (0 X ) , which maps (S1) labels to (S1) labels and maps(S2) labels to (S2) labels, except the case d = 1 is mapped to an (S1) label. Following asingle edge with vertex label ( Xs ) (here s = 0 , ) maps labels having s = 0 to (2 X ) ,which preserves the property of being an (S1) label or an (S2) label. For the case s = 2 , NTERSECTIONS OF MULTIPLICATIVE TRANSLATES OF -ADIC CANTOR SETS II: TWO INFINITE FAMILIES17 which must be an (S1) label, rewrite ( Xs ) = ( Y j ) for some j ≥ , which is convertedto (2 Y j − ) , which is an (S2) label. The extreme case ( Xs ) = (2 k ) is converted to m = 3 k , in (S2). This completes the induction step.(G2) There are clearly k elements in (S1). The reflection map R acts on elements m of (S1) with m > by replacing each by and vice versa, except that the smallest isconverted to a , and this is an element of (S2). The remaining element m = 0 exchangeswith m = 3 k which is in (S2). Conversely elements of (S2) are mapped into elements of(S1), for m < k an expression j is converted to j , and for m = 3 k is sent to m = 0 .Since the reflection map is an involution, it is one to one, so the (S2) labels have the samecardinality k as (S1) labels.(G3) This assertion is proved by induction on the length of the path. It is vacuously trueat step . For the induction step we must check that the vertices m and k − m have thesame number of exit edges, and that the available exit edges have reversed labels in thesecond case. We must also check that following an edge in the two cases leads to a pair ofreflected vertex labels m ′ and k − m ′ . There are several cases.Case (1) If m = ( X ℓ ) for ℓ > of type (S1), then k − m = ( ¯ X ℓ ) is of type (S2).Both allow , exit edges. A exit edge from m goes to m ′ = (0 X ℓ − ) , and a exit edge for k − m goes to (2 ¯ X ℓ − ) = 3 k − m ′ . A exit edge from m goesto m ′′ = (2 X ℓ − ) , and a exit edge for k − m goes to (0 ¯ X ℓ − ) = 3 k − m ′′ .Case (2) If m = ( X ℓ ) for ℓ > of type (S1), then k − m = ( ¯ X ℓ − is of type(S2). Here m allows only a exit edge, while k − m allows only a exit edge.Under the allowed exit edge m goes to m ′ = (2 X ℓ − ) of type (S2). Underthe allowed exit edge k − m goes to (0 ¯ X ℓ − ) = 3 k − m ′ of type (S1).For the two further cases where m is of type (S2), reverse the above. This completes theinduction step.(G4) By (G3) if a vertex labeled m is reachable from (0) , then its reflected vertex k − m is reachable from vertex k . But vertex k is reachable from (0) so k − m isreachable from (0) as well.(G5) We may assume that the (S1) vertex m = 0 , so it has the form r r r · · · r j ,in which all r i > except possibly r and r j , and r + r + · · · + r j = k . Nowit may be realized following a directed path from (0) having successive edge labels r j , r j − , r j − , · · · , r . This path is legal, because all intermediate words in the pathhave initial -adic digit so both edges labeled and exit from that vertex. (The intialword has k initial zeros, and each step can decrement the number of leading zeros by atmost ).(G6) One first checks that each label m in (S1) ending in corresponds under reflectionto a label k − m in (S2) ending in and vice versa (since divides m ). Each label in(S1) ending in corresponds under reflection to a label in (S2) ending in ; the (S1) labelpermits only a single exit edge with label and the corresponding (S2) label has a singleexit edge labeled . Thus at each vertex the reflection automorphism (at the level of vertexlabels) preserves the number of edges and reverses their edge labels. This establishes (G6).Moreover the graph G is a double cover of the quotient graph H under the automorphism R (which has no fixed points). (cid:3) Our next object is to show that the underlying graph G k of the path set X (1 , N k ) has atleast ⌈ k +12 ⌉ nested connected components, a number which is unbounded as k → ∞ . Weestablish this using the following notion of depth to vertices of G k . Definition 4.3. (1) First we classify the labels of the vertices in graph G k as being of types(T1) and (T2) as follows:(T1) The k -th -adic digit of m is or , so m = (0 X ) or m = (1 X ) , with X containing k − digits, but excluding the label m = 3 k = (10 k ) .(T2) The k -th -adic digit of m is , i.e. m = (2 X ) , as above, in addition includingthe label m = 3 k = (10 k ) .One may check that there are k elements in each set, and that the reflection operation R ( m ) = 3 k − m sends (T2) labels to (T1) labels and vice versa.(2) The depth of a (T1) label is the number of blocks of consecutive ’s appearing inits -adic expansion. The depth of a (T2) label m is the depth of its reflected label R ( m ) ,which is of type (T1).Thus m = 0 and m = 3 k are assigned depth . Furthermore all the vertices in the pathof length k + 2 studied in the proof of Theorem 9.1 are assigned depth , and they are thecomplete set of depth vertices.The following proposition will establish that this notion of depth stratifies the stronglyconnected components, by showing depth is nondecreasing along each directed edge. Proposition 4.4. For P k = 2 · k + 1 the path set X (1 , P k ) has presentation ( G k , v ) withthe following properties.(1) Each step along an edge in the graph G k leaves the same or increases the depth ofa vertex.(2) For ≤ j ≤ ⌊ k/ ⌋ there are exactly (cid:0) k +12 j +1 (cid:1) vertices in G k of depth exactly j .(3) For each ≤ j ≤ ⌊ k ⌋ , the vertices of depth j form a strongly connected componentof the underlying directed graph G k . Thus, G k has a sequence of ⌊ k/ ⌋ stronglyconnected components, which are nested in a chain.Proof. The presentation found in this theorem will be that given by the construction ofAlgorithm A in part I [3]. Some of the notation below only makes sense for k > . Wewill restrict to these cases, as the result follows for k = 1 , , by direct inspection. Thereversal operation exchanges type (T1) and type (T2) labels. For this to work the top -adicdigit (the k -th digit) must be used, because this is the only digit always reversed under thereflection map or with changed to ; there is one exception, which is m = 0 and m = 3 k ,where we assigned them to (T1) and (T2) directly. The key point is: a label m and itsreversal are always at the same level. For the two exceptions m = 0 and m = 3 k this facthad to be checked directly.(1) It suffices to check the effect of traversing a single edge in G k . The assertion holdsfor cases m = 0 and m = 3 k because they both exit to level vertices. By the proofof (G3) in Proposition 4.2, if label m goes to m ′ by edge labeled s , then k − m goes to k − m ′ by an edge labeled ¯ s . Now the depths of m and k − m are the same, as are thoseof m ′ and k − m ′ , so it suffices to check the effect of following an edge from a vertex oftype (T1). We treat cases.(i) Suppose m = (0 X of type (T1) has depth d , thus X contains d blocks ofconsecutive ’s. Following a edge goes to m ′ = (00 X ) , also (T1) of depth d .(ii) Suppose m = (0 X of type (T1) has depth d , thus it has d blocks of consecutive ’s. Following a edge goes to m ′ = (20 X ) , now (T2), of depth same as k − m ′ .Now X = X ′ ℓ with ℓ ≥ or X = 0 ℓ . In the first case k − m ′ = (02 ¯ X ′ ℓ ) If X ′ = 0 X ′′ , then it has d − blocks of ′ s , but its reversal ¯ X has d blocks.If X ′ = 2 X ′′ then it has d − blocks of ′ s , as does its reversal, but the at NTERSECTIONS OF MULTIPLICATIVE TRANSLATES OF -ADIC CANTOR SETS II: TWO INFINITE FAMILIES19 front creates another block. If X ′ = 0 X ′ then it has d blocks of ’s, as does itsreversal. Finally if X ′ = 2 X ′ then it has d blocks of ’s, its reversal has d − blocks, but the at front creats another blocks. In all cases the depth cannotdecrease.(iii) Suppose m = (0 X ℓ ) with ℓ > of type (T1) has depth d . Now can onlyfollow a edge, go to m ′ = (20 X ℓ − ) is of type (T2). This has same depth as k − m ′ = (02 ¯ X ℓ − ) . Now X has d − blocks of ’s. If it is of form X ′′ then reversal increases number of blocks of ’s in it by , compensating exactlyfor the lost block at the right end of the label, so the depth is still d . If of form X ′′ or X ′′ then reversal leaves d − blocks of ’s but get one extra blockfrom either before or after, so the depth is still d . If of form X ′′ then reversalleaves d − blocks of ’s but now gain two extra blocks from the before andafter, so the depth is still d .In all cases of a type (T1) vertex a step leaves depth the same or increases it by .(2) Let k be fixed. The result is true for j = 0 by the construction in Theorem 9.1, wherethere are k + 2 = 2 (cid:0) k +11 (cid:1) vertices of depth , and this component is strongly connected.For j ≥ it suffices to count the number of labels of type (T1) at depth j and thendouble it. For j ≥ the number of labels of type ( T at depth j consist of all labelsof form (0 k ℓ k ℓ · · · k j ℓ j k j +1 X ) with final block X = ∅ (set k j +2 = 0 ) or X = (10 k j +2 − ) (the latter requires k j +2 ≥ ). Since labels have length k the exponentsnecessarily satisfy k + · · · + k j +1 + k j +2 + ℓ + · · · + ℓ j = k, k i , ℓ i > for ≤ i ≤ j ; k j +1 , k j +2 ≥ . There are (cid:0) k j (cid:1) solutions of depth j type ( T with X not containing a ; this follows sincethere are k symblols in a label and we mark the final elements of each k i and k i withan asterisk for ≤ i ≤ j to uniquely determine a depth j label with X = ∅ . There are (cid:0) k j +1 (cid:1) solutions of depth j type ( T with X containing a ; here we add an additionalasterisk marking the , which unqiuely specifies the label, so we have the number of waysof inserting j + 1 asterisks. Thus the number of ( T labels of depth j is (cid:0) k +12 j +1 (cid:1) , and (2)follows.(3) First, we show that it is possible to reach a vertex of each depth ≤ j ≤ ⌊ k/ ⌋ .Starting from m = 0 following paths with labels (10) j for ≤ j ≤ ⌊ k/ ⌋ , one arrives atvertices m j := ((02) j k − j ) , and m j is a type (T1) label of depth j . These are legalpaths since all the intermediate vertex m j labels (for ≤ j ≤ m − ) have initial -adicdigit . We have produced a path with vertices of depth , , , ..., ⌊ k/ ⌋ , which guaranteesthe existence of at least one sequence of distinct strongly connected components of length ⌊ k/ ⌋ which are nested in a chain.Next, we show that the subgraph of G k consisting of those vertices of depth j is stronglyconnected for each ≤ j ≤ ⌊ k/ ⌋ . At depth d = 0 , beginning at the vertext labeled andtraversing a path with label k +1 k +1 gives a loop at the -vertex that passes through eachother vertex of depth , so the subgraph of depth vertices is strongly connected.. Below,we restrict attention to depths d ≥ , and some statements below only apply in those cases.Recall also that we are restricting attention to k > , as smaller cases can be checked byhand.We need to show, firstly, that from any vertex it is always possible to traverse an edgethat leaves the depth unchanged. By the proof of (G3) in Proposition 4.2 and the discussionin the first paragraph of (1) above, it suffices to verify this for vertices of type (T1). Let m be the label of a vertex of depth d and type (T1). Then either m = (0 X , in which case we may follow an edge labeled to arrive at a vertex labeled (00 X ) that also has depth d ,or else m = (0 X l ) for some l > . In the latter case, we may follow an edge labeled to a vertex labeled (20 X l − ) , and the discussion in (iii) above shows that this vertexalso has depth d . In any case, we can always traverse an edge that will leave the depthunchanged.Among depth d labels, the minimal such label is m min = ((20) d − . In order toshow that the set of depth d vertices is a strongly connected subgraph of G k , it sufficesto show that it is always possible, beginning at any vertex of depth d , to traverse pathsboth forwards to m min and backwards to the same vertex (that is, contrary to the ordinarydirection that arrows are traversed; this will show that there is a path forwards from m min to the desired vertex). This will follow if we can show that:(A) For any depth d vertex with non-minimal label m , it is always possible to follow apath, staying at depth d , to another vertex with label m ′ < m .(B) For any depth d vertex, it is possible to follow edges backwards until we reach avertex where each block of ’s has length exactly .(C) For any depth d vertex with a label where each block of ’s has length exactly , it ispossible to reach m min by going backwards.(A) Suppose now we are at a depth d vertex with label m of type (T1). Then either m is of the form (0 X , or else m is of the form (0 X l ) for some l > . If m = (0 X ,then we may traverse an edge labeled to arrive at an edge labeled m ′ = (0 X ) < m , and m ′ is also at depth d . Now suppose instead that m = (0 X l ) . Then we must traversenext an edge labeled to the vertex with label m ′ = (20 X l − ) > m . By the argumentof (iii) above, this vertex also has depth d . From here, we may traverse l consecutive edgeslabeled to arrive at a vertex labeled m ′′ = (20 X ) , whose depth is also d . If the right-most digit of X is not a , we may continue to traverse edges labeled until we arrive ata vertex m ′′′ = (20 Y ) where the right-most digit of Y is a , and the length | Y | ≤ | X | ,or else at the vertex m (4) = (2) if X is the empty string. In the latter case, we are atdepth d = 1 and m (4) = (2) = m min is already the minimal label. Suppose we are inthe former case, and we have arrived at m ′′′ = (20 Y ) . But for any l ≥ , we necessarilyhave m ′′′ = (20 Y ) ≤ ( X l ) = m , with equality if and only if X = Y , l = 1 ,and m = m ′ = (20) d − m min . Thus, in any case, we may always traverse a path,remaining at depth d , to arrive at a vertex whose label is less than m .What if our initial vertex is of type (T2)? Then, m is either of the form k , in whichcase, we simply follow edges labeled until we reach the vertex labeled , or we havesomething of the form X , where X has k − digits. In this case, if X terminates in l ,we can immediately follow a vertex , without dropping depths, to m ′ of form ( T , whereof course m ′ < m . Otherwise, we have Y l , where we follow l + 1 edges of label ;the first l bring us to Z , and the ( l + 1) st edge takes us to a (T2) vertex that terminatesin n , which is a case already covered.This proves (A).To see (B), we will devise an algorithm (call it Algorithm (B)).(i) If we are at X l then we follow a vertex labeled 1 backwards to vertex X l +1 .(This does not drop depth, as a block of consecutive ’s necessarily transforms intoanother block of consecutive ’s).(ii) If we are at l X , where l > , or we are at l Y n , where l > , we follow a vertexlabelled to l − X or l − Y n +1 .(iii) If we are at X , and X omits the digit , we follow an edge labeled back to X .Notice that this avoids dropping depth. NTERSECTIONS OF MULTIPLICATIVE TRANSLATES OF -ADIC CANTOR SETS II: TWO INFINITE FAMILIES21 (iv) If we are X , where X omits the digit , we follow an edge labeled back to X .The crux is step (iii); following the notation of that step, we will then be at X , withno s after the . We then apply case (i), reaching X . Any other ’s that appeared in theblock at the far left will be transformed into ’s on the far right by the application of step ( iv ) , while the other blocks will merely be shifted.Thereby, by repeated application of this algorithm, all of the blocks will be transformedinto single-digit blocks after at most k iterations. This concludes (B). For an illustration atdepth 2, see the column labeled “Step (B)" in Table 4.1.Finally, for (C), notice that, for the type of vertex we are interested in, repeated applica-tion of Algorithm (B) simply "scrolls through" the label, with the blocks of ’s shifting left,always preserving the same cyclic order, with the same gaps of ’s between them (unlessa is present) between them. In the case of the illustration of Table 4.1, see the columnlabeled “Step (C)-1" of that table.So, for (C), apply Algorithm (B) until we are at l X where l > (if this is strictly im-possible, then simply "scroll" until we are at (02) k/ , and at this depth, that is the minimalvertex). Then, break the pattern and go to l X . Then, continue to apply Algorithm (B)until we return to a vertex where all of the blocks of ’s have length .Essentially, we will generate a long block of ’s instead of the block of ’s we currentlyhave, which won’t have such a large gap; see the column labeled “Step (C)-2" in Table 4.1.One such procedure transforms a block of ’s of arbitrary length into a block of length . Repeat this procedure untill all of the blocks of ’s (except for 1) have length , andthen use Algorithm (B) until we reach the minimal vertex. This completes (3). Continuingwith our simple example, see the column labeled “Step (C)-3" in Table 4.1.Step (B) Step (C)-1 Step (C)-2 Step (C)-322022022 0020002 0020002 000202020220220 0200020 0200021 002020002202200 2000201 2000210 020200022002201 0002002 0002022 202000120022002 0020020 0020220 020000200220020 0200200 0202200 200002102200200 2002001 2022001 000020222002001 0020002 022000220020002 220002100200020 20002020002020TABLE 4.1. Example of algorithm for proof of Proposition 4.3(3). (cid:3) Remark . (1) Proposition 4.4 counts the number of vertices at each depth, giving a re-cursion to compute them. Table 4.2 below gives values for ≤ k ≤ . Depth= P = 7 4 P = 19 6 2 P = 55 8 8 P = 163 10 20 2 P = 487 12 40 12 P = 1459 14 70 42 2 P = 4375 16 112 112 16 P = 13123 18 168 252 72 2 P = 39367 20 240 504 240 20 TABLE 4.2. Number of vertices at given depth in graph G k for X (1 , P k ) .(2) Proposition 4.4 says that the graph X (1 , P k ) has a “Matryoshka doll" structure of asingle set of nested strongly connected components, one at each depth ≤ j ≤ ⌊ k/ ⌋ .(3) The proof of Proposition 4.4 exploits repeatedly the symmetry of the graph G k exhibited by the partitioning of vertices into types (T1) and (T2).4.2. The Family P k = (20 k − = 2 · k + 1 : Hausdorff dimension. Data on theHausdorff dimensions of the first few of the sets C (1 , P k ) were obtained by computercalculation of the maximum eigenvalue of the adjacency matrix of the graph X (1 , P k ) and presented in Section 3.1. The data contained oscillations and other features which wediscuss in Remark 4.6 below.We now lower bound the Hausdorff dimension of C (1 , P k ) as k → ∞ . Theorem 2.2gives both an asymptotic limiting result and a lower bound because it may be that theHausdorff dimensions continue to oscillate for large k . Proof of Theorem 2.2. Let a = ⌊ k ⌋ and let b ∈ { , , , } be congruent to k mod , sothat k = 4 a + b . Let S ⊂ A N = { , , } N be given by S = { (1100) a b ((1 x a b (1000) a − b ) ∞ ∈ A N | x ∈ { , } may vary } . (4.1)What we will show is that S ⊂ X (1 , P k ) . Since elements of S , after the fixed initial string (1100) a b , consists of symbol sequences of length k − with k − − a fixed digits and a digits which may be either or , it follows that H top ( S ) = a k − (2) = ⌊ k ⌋ k − (2) . The two inequalities of the theorem, that lim inf k →∞ dim H C (1 , P k ) ≥ 18 log (2) , and, for all k , dim H ( C (1 , P k )) ≥ 113 log (2) , then will follow immediately.To prove that S ⊂ X (1 , P k ) , we will trace out paths on the graph presentation of C (1 , P k ) given by Algorithm A of [3] whose edge labels give the elements of S . First,note that if we begin with an edge labeled from the -vertex, we arrive at the vertex withlabel k − . This means that our next k − vertices may be either or freely. Eachedge appends a to the front of the vertex label and removes the last digit, and eachedge appends a to the front of the vertex label and removes the last digit. From these NTERSECTIONS OF MULTIPLICATIVE TRANSLATES OF -ADIC CANTOR SETS II: TWO INFINITE FAMILIES23 observations, we see that there is in fact a sequence of edges with label (1100) a b , andhaving traversed these edges we arrive at a vertex labeled b (0022) a . Call this vertex v .We will now show that we may traverse a sequence of edges with label (1 x a b (1000) a − b initiating at v for x = 0 and x = 1 , and that such a path alsoterminates at v . The result will follow. Now since the label of v ends in , the only outedge is indeed labeled , and this takes us to a vertex labeled b (0022) a − . The nextedge label x may then be either of , terminating in a vertex labeled [2 x ]20 b (0022) a − ,where [2 x ] is a digit given by the product of and x . From this vertex we may traverse twosubsequent edges each labeled , and the target vertex is x ]20 b (0022) a − . It is easy tosee that we may repeat this process, traversing edges labeled (1 x a times and ultimatelyterminating at a vertex labeled (00[2 x ]2) a b . Traversing then b edges labeled gets us tothe vertex labeled b (00[2 x ]2) a . We may then traverse edges labeled (1000) a − b toarrive back at the vertex v labeled b (0022) a . This completes the proof. (cid:3) Remark . We speculate on the behavior of the Hausdorff dimension function C (1 , P k ) as a function of k . We believe the following might be true.(1) Fixing level j and varying k the topological entropy of the strongly connectedcomponent at depth j stay at value until k ≥ j − , then increas monotonicallyto a maximum and then decrease monotonically thereafter.(2) The “champion" depth j with maximal topological entropy is a nondecreasingfunction of k .Speculations (1) and (2) are suggested by analogy with the behavior of the number ofvertices at depth j as a function of k , given in Table 4.1, which have both these properties.4.3. Hausdorff dimension bounds for C (1 , P k , ..., P k n ) . The path set structures of themembers of the infinite family P k are compatible with each other, as a function of k , sothat the associated C (1 , P k , ..., P k n ) all have positive Hausdorff dimension. We relatethese Hausdorff dimensions to those of the infinite family L k = (1 k ) = (3 k +1 − treated by the first and third authors in [3] and reviewed in Appendix A (Section 8). Theorem 4.7. For the family P k = 2 · k + 1 = (20 k − , and ≤ k < . . . The graphs under consideration are the graphs given by Algorithm A of [3]. Sincethe underlying graph G k of the path set presentation ( G k , v ) of the path set X (1 , P k ) contains a double covering of the underlying graph G ′ k +1 of the path set presentation of X (1 , L k +1 ) , and G (1 k ) ⋆ · · · ⋆ G (1 kn +2 ) ∼ = G (1 kn +2 ) , the proposition follows from Theorem 9.1 in Appendix B.Note that this directed graph covering is not a covering at the level of path sets, becausethe path labels on the two graphs differ. (cid:3) Theorem 4.7 shows that there exist an arbitrarily large number of different values M j ,each having a in their ternary expansion, such that dim H ( C (1 , M , M , ..., M n )) > . 5. T HE INFINITE FAMILY Q k = 3 k − k + 1 = (2 k k − Let Q k = 3 k − k + 1 = (2 k k − . We will prove Theorem 2.3, which describesthe structure of a graph presentation G k of C (1 , Q k ) . We then use this description to proveTheorem 2.4, which computes the Hausdorff dimension of C (1 , Q k ) .5.1. The Family Q k = (2 k k − = 3 k − k + 1 : Path set structure. First, let us givean example. The following example gives a path set presentation for Q = 73 . Example 5.1. A path set presentation of X (1 , , with 73 = (2201) , is shown in Fig-ure 5.1. The vertex labeled is the marked initial vertex.The graph in Figure 5.1 has adjacency matrix A = , which has Perron eigenvalue β = √ , so dim H ( C (1 , √ ! ≈ . . We describe the path set presentation in general. Theorem 2.3 will follow easily fromthe following result, which makes use of the concepts developed in Section 3.4. Proposition 5.1. Let P = X (1 , be the path set underlying C (1 , , and let Q = X (1 , Q k ) be the path set underlying C (1 , Q k ) . Then Q is the interleaved path set Q = P ( ∗ k ) . (5.1) Proof. For convenience, we recall that P = X G (0) for the graph G in Figure 5.1. This isthe graph given by the Algorithm A of [3].Let ( H , v ) be the graph presentation of Q given by the same algorithm. An elementof P may begin with either a or a , while an element ( x i ) ∞ i =0 of Q may begin withany sequence x x · · · x k − of ’s and ’s, since Q k terminates in k − . Thus, the initial k -blocks of Q are precisely the same as the initial k -blocks of the interleaved path set P ( ∗ k ) .To show that Q = P ( ∗ k ) we just need to check that for each ≤ j ≤ k − , theadmissible strings x j x j + k x j +2 k · · · of j (mod k ) digits of elements of Q are preciselythe elements of P . We proceed by induction on j ≥ , the observation above complet-ing the base case j = 0 . Inductively, assume none of the digits x r for r ≡ l (mod k ) NTERSECTIONS OF MULTIPLICATIVE TRANSLATES OF -ADIC CANTOR SETS II: TWO INFINITE FAMILIES25 011 01 1 1 0 01 000 11 00 01011 01FIGURE 5.1 Path set presentation of X (1 , . The marked vertex is .with l < j can restrict the admissible values for the digits x j + nk for n ≥ . We mean X (1 , . The marked vertex is .here that whether x r = 0 or x r = 1 has no effect on the last digit of the vertex la-bel in H arrived at from a path labeled x x · · · x j + nk originating at v . The base case, j = 0 , is satisfied trivially. Then we can without loss of generality assume x i = 0 forall ≤ i < j . For now, we will also assume that x r = 0 for all r j (mod k ) .This assumption is not as restrictive as it seems since, as we will show, the j (mod k ) digits do not effect the available choices for digits of other modular classes. Now since Q k = 2 k k − , whether x j is or has no effect on the digits x j +1 , x j +2 , . . . , x j + k − .If x j = 0 , then x j + k may also be either or . If x j + mk is for all m < n , thenalso x j + nk may be either or , and those x r for r < j + nk , r j (mod k ) are un-restricted. On the other hand, suppose there is an n ≥ such that x j + mk = 0 for all m < n and x j + nk = 1 . Again, the labels x r for r < j + ( n + 1) k , r j (mod k ) areunrestricted. However, x j +( n +1) k must now be a . Now the label of the vertex we are at,having traversed the path labeled x x · · · x j +( n +1) k from v , has label k − . Thus thedigits x j +( n +1) k +1 , x j +( n +1) k +2 , · · · x j +( n +3) k − are unrestricted. However, if the digit x j +( n +2) k is a , then the vertex at the end of the path labeled x x · · · x j +( n +2) k has label k − , so the vertices after x j +( n +2) k are restricted or unrestricted in precisely the sameway as those after x j +( n +1) k . If on the other hand x j +( n +2) k = 0 , then the terminal vertexhas label k − . Thus, the label of the vertex after j + ( n + 3) k − steps in this caseis , hence in this case x j +( n +3) k must be . The resulting terminal vertex label is . Ineither case, the digits, x j +( n +3) k +1 , x j +( n +3) k +2 , x j +( n +4) k − are unrestricted. For the ( j + ( n + 4) k ) th step we either begin at vertex or at vertex k − , which cases havealready been considered.Thus, we have shown that the digits x j + nk place no restrictions on any digits fromthe other modular classes, and, furthermore, we have described the restrictions that x j + nk place on x j + mk for m > n . Inspecting this description shows that the admissible digits x j x j + k x j +2 k are precisely the edge labels of the infinite walks in G originating at thevertex in Figure 5.1. These are precisely the elements of P , so Q = P ( ∗ k ) . (cid:3) NTERSECTIONS OF MULTIPLICATIVE TRANSLATES OF -ADIC CANTOR SETS II: TWO INFINITE FAMILIES27 Let G be the graph of Figure 5.1. The presentation for Q k given by Proposition 3.4applied to k and G is isomorphic to that given by Algorithm A of [3]. We are now ready toprove Theorem 2.3. Proof of Theorem 2.3. Let ( G k , v ) be the presentation of Q = X (1 , Q k ) constructed byapplying the algorithm of Proposition 3.4 to the presentation G of X (1 , . Since the graph G used in this construction has vertices and edges, it follows by Proposition 3.4 that G k has k vertices and · k − edges. Moreover, since G is strongly connected, so is G k , byRemark 3.5. This proves the theorem. (cid:3) The family Q k = (2 k k − = 3 k − k + 1 : Hausdorff dimension. We haveshown that X (1 , Q k ) = X (1 , ( ∗ k ) , (5.2)is given by an interleaving construction. Using the results of Section 3.4, it is now a simplematter to prove Theorem 2.4. Proof of Theorem 2.4. We are trying to show that dim H ( C (1 , Q k )) = log φ. The result follows by Proposition 5.1 and by application of the interleaving result given inCorollary 3.7, since dim H ( C (1 , φ, as is easily computed, and Corollary 3.7 shows that the interleaving operation ( · ) ( ∗ k ) pre-serves the topological entropy of the input path set. (cid:3) 6. B OUNDS ON H AUSDORFF DIMENSIONS BY NUMBERS OF TERNARY DIGITS We study properties of the Hausdorff dimension constants α n .6.1. Upper Bound on Γ via n -digit constants α n : Proof of Theorem 2.5. It is knownthat the number of nonzero ternary digits in (2 n ) goes to infinity as n → ∞ , i.e. for each k ≥ there are only finitely many n with (2 n ) having at most k nonzero ternary digits.This result was first established in 1971 by Senge and Straus, see [19]. In 1980 Colin L.Stewart [21, Theorem 1] obtained a quantitative refinement of such bounds. We obtain asa special case of his result the following quantitative version of the rate of growth of thenumber of nonzero digits. Theorem 6.1. (C. L. Stewart) For each k ≥ , there are only finitely many n such that thebase expansion of n (equivalently the -adic expansion (2 n ) ) has at most k nonzerodigits. More precisely, if n ( n ) denotes the sum of the base digits of n , then for m ≥ , n (2 m ) > log m log log m + c − , where c > is an effectively computable constant.Proof. The result follows from [21, Theorem 1], taking for bases a = 2 , b = 3 , and digits α = β = 0 . Using Stewart’s notation, L a,α (2 m ) = 2 , so that L a,α,b,β (2 m ) − counts thenumber of nonzero ternary digits n (2 m ) of m . (cid:3) We can now prove Theorem 2.5. Proof of Theorem 2.5. For each n ≥ we have Γ ≤ dim H ( E ( n +1)1 ) . We also have the inclusions E ( n +1)1 = [ ≤ m <... For all M ≥ with M ≡ , one has dim H ( C (1 , M )) ≤ log φ ≈ . . where φ = √ is the golden ratio. Thus α = log φ ≈ . Proof. We may write M = ( m n m n − . . . m k k − for some ≤ k ≤ n < ∞ since M is an integer, M ≡ . Our strategy will be to construct an injective map f : C (1 , M ) → C (1 , N k ) , where recall that N k = (10 k − , and by [3, Theorem 1.8], dim H ( C (1 , N k )) = log ( φ ) . Let ( G , v ) and ( H k , w ) be the right-resolving, connected,essential presentations of C (1 , M ) and C (1 , N k ) , respectively, constructed by Algorithm Aof [3]. The injective map f induces for each l an injective map from the set of paths oflength l in G originating at v to the set of paths of length l in H k originating at w , sincethere is a bijective correspondence between elements of C (1 , M ) or C (1 , N k ) and infinitepaths in G or H k , respectively, originating at the distinguished vertex. Thus, following [1,Definition 1.10] and [2, Theorem 1.1], this will establish the result.To define the map f : C (1 , M ) → C (1 , N k ) , we will need some notation. Let α = . . . a a a be a generic element of C (1 , M ) . α corresponds to a vertex path . . . v v v of G such that there is an edge labeled a i from vertex v i to vertex v i +1 . We call the digit a i NTERSECTIONS OF MULTIPLICATIVE TRANSLATES OF -ADIC CANTOR SETS II: TWO INFINITE FAMILIES29 restricted if the out-degree of v i is , and we call a i unrestricted if the out-degree of v i is . We call a i restricting if a i + k is restricted, and otherwise we call a i non-restricting .If the digit a i of α is unrestricted, then it is possible to find an element α ′ = . . . a i + k − a i + k − . . . a i +1 (1 − a i ) a i − . . . a a a ∈ C (1 , M ) . That is, changing a i to − a i does not require us to make any other changes until the i + k -th digit. Thenfor all such α ′ the vertex v ′ i + k of the corresponding vertex path on G is the same. If a i is not only unrestricted but also restricting, then if this vertex v ′ i + k has out-degree , wecall a i unconditionally restricting , and if v ′ i + k has out-degree , we call a i conditionallyrestricting . Thus, a conditionally restricting digit can be changed to become unrestricting,while an unconditionally restricting digit remains restricting when changed.Tautologically, a conditionally restricting digit a i becomes unrestricting when replacedby − a i , but we can also see that an unrestricted, unrestricting digit a i becomes condi-tionally restricting when replaced by − a i , since this necessarily changes the carry digitat the ( i + k ) -th step. Thus, these types of digits come in pairs.Now we are ready to construct the map f : C (1 , M ) → C (1 , N k ) , digit-by-digit, for α ∈ C (1 , M ) : f ( α ) i = if a i is restricted or unrestricting ; a i if a i is unrestricted and unconditionally restricting ;1 if a i is unrestricted and conditionally restricting . (6.3)Though f ( α ) is clearly an element of Σ , we need to check first that it is really anelement of C (1 , N k ) . To see this, note that if f ( α ) i = 1 , then a i was restricting, so a i + k is restricted, thus f ( α ) i + k = 0 . So a digit of f ( α ) is always followed, k digitslater, by a digit . Since C (1 , N k ) can be described as the Z / Z -shift of finite type withforbidden block set { k − } , and this block does not occur in f ( α ) , we are assured that f ( α ) ∈ C (1 , N k ) .It remains only to check that f is injective. Suppose α = . . . a a a , β = . . . b b b ∈C (1 , M ) are distinct. Then there is a j such that a j = 1 − b j and a i = b i for all ≤ i < j . Let . . . v v v and . . . w w w be the vertex paths of G corresponding to α and β , respectively. Then we must have v i = w i for ≤ i ≤ j , and v j = w j must haveout-degree . Thus, the digits a j of α and b j of β are unrestricted. But by the discussionabove, if a j is conditionally restricting then b j is unrestricting, in which case f ( α ) j = 1 =0 = f ( β ) j , and vice versa, or else a j and b j are both unconditionally restricting, in whichcase f ( α ) j = a j = b j = f ( β ) j . In any case, we see that f ( α ) = f ( β ) , so f is injective,establishing the result. (cid:3) 7. B LOCK NUMBER AND INTERMITTENCY OF TERNARY EXPANSIONS The examples given so far show that the dependence of dim H ( C (1 , M )) for a posi-tive integer M is complicated function, being driven by the structure of the underlyingautomata, whose construction includes aspects of both number theory and dynamical sys-tems. One may ask whether the Hausdorff dimension might go to zero as a function ofsome statistic easily computable from the ternary expansion ( M ) . Earlier results of thispaper show that the statistic d ( M ) does not have this property.We now present empirical results for two other interesting statistics of ( M ) :(1) The block number b ( M ) counts the number of blocks of consecutive nonzerodigits in the ternary expansion ( M ) .(2) The intermittency s ( M ) counts the number of distinct blocks of consecutivematching digits in the ternary expansion ( M ) . We clearly have b ( M ) ≤ s ( M ) . As examples, b ((2121011) ) = 2; b ((2101) ) = 2 , while s ((2121011) ) = 6; s ((2101) ) = 4 . The statistic b ( M ) might be relevant to controlling the Hausdoff dimension since blocksof zeros at the end of the number have a simple effect on the associated automaton.Table 7.1 below presents data on Hausdorff dimensions for a few numbers M taking thesmallest values for s ( M ) , computed using the algorithm in Part I to six decimal places.The table also provides the number of vertices in the associated finite directed graph.Path Set C (1 , M ) ( M ) s ( M ) Vertices Perron eigenvalue Hausdorff dim C (1 , 10) 101 3 4 1 . . C (1 , 16) 121 3 5 1 . . C (1 , 19) 201 3 8 1 . . C (1 , 73) 2201 3 16 1 . . C (1 , 34) 1021 4 8 1 . . C (1 , 46) 1201 4 10 1 . . C (1 , 61) 2021 4 14 1 . . C (1 , 64) 2101 4 14 1 . . C (1 , 70) 2121 4 14 1 . . C (1 , 91) 10101 5 9 1 . . C (1 , 97) 10121 5 16 1 . . C (1 , . . C (1 , . . C (1 , . . C (1 , . . C (1 , . . C (1 , . . C (1 , . . C (1 , . . C (1 , . . TABLE 7.1. Hausdorff dimension of C (1 , M ) by intermittencyThis extremely limited data set exhibits a small decrease in Hausdorff dimensions as thestatistic s ( M ) increases. It leaves open the possibility that one might have dim H ( C (1 , M )) → as b ( M ) → ∞ , noting that b ( M ) ≤ s ( M ) . Further numerical experimentation seemswarranted to get a better idea whether such an assertion might be true.Regarding potential applicability of information on these statistics to the Exceptionalset conjecture, we must point out that it is not currently known whether b (2 n ) → ∞ holds as n → ∞ or whether s (2 n ) → ∞ holds as n → ∞ . 8. A PPENDIX A: R EVIEW OF RESULTS FOR FAMILIES L k = (1 k ) AND N k = (10 k − .We review two results proved in [3, Section 4]. The first is for the family L k = (3 k − 1) = (1 k ) , for k ≥ , given as [3, Theorem 5.2]. NTERSECTIONS OF MULTIPLICATIVE TRANSLATES OF -ADIC CANTOR SETS II: TWO INFINITE FAMILIES31 Theorem 8.1. (Infinite Family L k = (3 k − ) (1) Let L k = (3 k − 1) = (1 k ) . The path set presentation ( G , v ) for the path set X (1 , L k ) underlying C (1 , L k ) has exactly k vertices and is strongly connected.(2) For every k ≥ , dim H ( C (1 , L k )) = dim H C (1 , (1 k ) ) = log β k , where β k is the unique real root greater than of λ k − λ k − − .(3) For all k ≥ there holds dim H (cid:16) C (1 , L k ) (cid:17) = log kk + O (cid:18) log log( k ) k (cid:19) . The Hausdorff dimension dim H ( C (1 , L k )) is positive but approaches as k → ∞ . Wepresent data in Table 8.1 below.Path set L k Vertices Perron eigenvalue Hausdorff dim C (1 , L ) . . C (1 , L ) . . C (1 , L ) 13 3 . . C (1 , L ) 40 4 . . C (1 , L ) 121 5 . . C (1 , L ) 364 6 . . C (1 , L ) . . C (1 , L ) . . C (1 , L ) . . TABLE 8.1. Hausdorff dimensions of C (1 , L k ) (to six decimal places)We also recall results on the family N k = 3 k + 1 = (10 k − , which consists ofnumbers with exactly two nonzero ternary digits, with s ( N k ) = 2 , given as [3, Theorem5.5]. Theorem 8.2. (Infinite Family N k = 3 k + 1 ) (1) Let N k = 3 k + 1 = (10 k − . The path set presentation ( G , v ) for the path set X (1 , N k ) underlying C (1 , N k ) has exactly k vertices and is strongly connected.(2) For every integer k ≥ , there holds dim H ( C (1 , N k )) = dim H C (1 , (10 k − ) = log (cid:18) √ (cid:19) ≈ . . Here the Hausdorff dimension is constant as k → ∞ .9. A PPENDIX B: R ELATION OF FAMILIES P k = (20 k − AND L k +1 = (1 k +1 ) We observe a relation between the Hausdorff dimensions of C (1 , P k ) and C (1 , L k +1 ) .For ≤ k ≤ , the Hausdorff dimension of C (1 , (20 k − ) equals that of C (1 , (1 k +1 ) ) .For general k we obtain an inequality. Theorem 9.1. The Hausdorff dimensions of C (1 , P k ) and C (1 , L k +1 ) are related by dim H ( C (1 , P k )) ≥ dim H ( C (1 , L k +1 )) . (9.1) Proof. The marked vertex v with label (0) of the path set presentation G (20 k − associ-ated to C (1 , (20 k − ) has two exit edges, one a self-loop with edge labeled , the second an exit edge labeled to the vertex labeled (20 k − ) . From this vertex, there is an edgelabeled to the vertex labeled (220 k − ) . This continues for k − more steps into a vertexlabeled (2 k ) , from which there is an out-edge labeled to a vertex labeled (10 k ) . Thereis a self-loop labeled at the (10 k ) -vertex, and a path of length k + 1 through vertices (10 k − j ) , for ≤ j ≤ k , all with edge label , then back to the -vertex. Considering onlythe edges given above, this comprises a subgraph H of G (20 k − having k + 2 edgesthat is strongly connected, and consists of a closed path starting and ending at of length k + 2 plus two self-loops, at vertices m = 0 and m = 3 k . (The case k = 2 is pictured inExample 4.1, where the subgraph of G (201) under consideration is the six outer vertices inthe graph in Figure 4.1.) Upon inspection we see that the graph H is a double-covering ofthe graph G (1 k +1 ) associated to C (1 , L k +1 ) given by Algorithm A of [3]. This implies thebound (9.1). (cid:3) Remark . For ≤ k ≤ , equality holds in Proposition 9.1 because the subgraphof G (20 k − constructed in the proof is the strongly connected component with great-est topological entropy in these cases. This is not true for almost all larger k . Theorem8.1 says dim H ( C (1 , L k )) → as n → ∞ . On the other hand Theorem 2.2 says that dim H ( C (1 , L k )) is bounded away from as k → ∞ .R EFERENCES[1] W. Abram and J. C. Lagarias, Path sets in one-sided symbolic dynamics, Advances in Applied Mathematics, (2014), pp. 109-134.[2] W. Abram and J. C. Lagarias, p -Adic path set fractals and arithmetic, Journal of Fractal Geometry, (2014),no.1, 45-81.[3] W. Abram and J. C. 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(1988), 483–490.D EPARTMENT OF M ATHEMATICS , H ILLSDALE C OLLEGE , H ILLSDALE , MI 49242-1205, USA E-mail address : [email protected] C OLLEGE OF THE S CHOOL OF N ATURAL S CIENCES AND M ATHEMATICS , U NIVERSITY OF T EXASAT D ALLAS , R ICHARDSON , TX 75080-3021, USA E-mail address : [email protected] D EPARTMENT OF M ATHEMATICS , U NIVERSITY OF M ICHIGAN , A NN A RBOR , MI 48109-1043,USA E-mail address ::