DIGIT FREQUENCIES AND SELF-AFFINE SETS WITH NON-EMPTY INTERIOR SIMON BAKER 7 Abstract. In this paper we study digit frequencies in the setting of expansions in non- 1 integer bases, and self-affine sets with non-empty interior. 0 Within expansions in non-integer bases we show that if β ∈ (1,1.787...) then every 2 √ x ∈ (0, 1 ) has a simply normal β-expansion. We also prove that if β ∈ (1,1+ 5) then n β−1 2 a every x ∈ (0, 1 ) has a β-expansion for which the digit frequency does not exist, and β−1 J a β-expansion with limiting frequency of zeros p, where p is any real number sufficiently 4 close to 1/2. 2 For a class of planar self-affine sets we show that if the horizontal contraction lies in ] S a certain parameter space and the vertical contractions are sufficiently close to 1, then D every nontrivial vertical fibre contains an interval. Our approach lends itself to explicit h. calculation and give rise to new examples of self-affine sets with non-empty interior. One t particular strength of our approach is that it allows for different rates of contraction in a m the vertical direction. [ 1 v 3 1. Introduction 7 7 Let x ∈ [0,1]. A sequence ((cid:15) ) ∈ {0,1}N is called a binary expansion of x if 6 i 0 ∞ 1. x = (cid:88) (cid:15)i. 0 2i 7 i=1 1 It is well known that apart from the dyadic rationals (numbers of the form p/2n) every : v x ∈ [0,1] has a unique binary expansion. The exceptional dyadic rationals have precisely i X two binary expansions. A seemingly innocuous generalisation of these representations is r a to replace the base 2 with a parameter β ∈ (1,2). That is, given x ∈ R, we call a sequence ((cid:15) ) ∈ {0,1}N a β-expansion of x if i ∞ (cid:88) (cid:15) i x = π (((cid:15) )) := . β i βi i=1 Date: 25th January 2017. 2010 Mathematics Subject Classification. Primary 11A63; Secondary 28A80, 11K55. Key words and phrases. Expansions in non-integer bases, Digit frequencies, Self-affine sets. 1 2 SIMON BAKER These representations were first introduced in the papers of Parry [28] and Rényi [29]. It is straightforward to show that x has a β-expansion if and only if x ∈ [0, 1 ]. In what β−1 follows we will let I := [0, 1 ]. β β−1 Despite being a simple generalisation of binary expansions, β-expansions exhibit far more exotic behaviour. In particular, one feature of β-expansions that makes them an interesting object to study, is that an x ∈ I may have many β-expansions. In fact a result β of Sidorov [30] states that for any β ∈ (1,2), Lebesgue almost every x ∈ I has a continuum β of β-expansions. Moreover, for any k ∈ N∪{ℵ }, there exists β ∈ (1,2) and x ∈ I such 0 β that x has precisely k β-expansions, see [7, 9, 15, 16, 31]. Note that the endpoints of I β always have a unique β-expansion for any β ∈ (1,2). A particularly useful technique for studying both binary expansions and β-expansions is to associate a dynamical system to the base. One can then often reinterpret a problem in terms of a property of the dynamical system. The underlying geometry of the dynamical system can then make a problem much more tractable. In this paper we prove results relating to digit frequencies and self-affine sets. These results are of independent interest but also demonstrate the strength of the dynamical approach to β-expansions. 2. Statement of results 2.1. Digit frequencies. Let ((cid:15) ) ∈ {0,1}N. We define the frequency of zeros of ((cid:15) ) to be i i the limit #{1 ≤ i ≤ n : (cid:15) = 0} i freq ((cid:15) ) := lim . 0 i n→∞ n Assumingthelimitexists. Wecallasequence((cid:15) )simply normal iffreq ((cid:15) ) = 1/2.Foreach i 0 i x ∈ [0,1], we let freq (x) denote the frequency of zeros in its binary expansion whenever 0 thelimitexists. Whenthelimitdoesnotexistwesayfreq (x)doesnotexist. Thefollowing 0 results are well known: (1) Lebesgue almost every x ∈ [0,1] has a simply normal binary expansion. (2) dim ({x : freq (x) does not exist}) = 1. H 0 (3) For each p ∈ [0,1] we have −plogp−(1−p)log(1−p) dim ({x : freq (x) = p}) = . H 0 log2 In (3) we have adopted the convention 0log0 = 0. The first statement is a consequence of Borel’s normal number theorem [10], the second statement appears to be folklore, and the third statement is a result of Eggleston [14]. DIGIT FREQUENCIES AND SELF-AFFINE SETS WITH NON-EMPTY INTERIOR 3 The above results provide part of the motivation for the present work. In particular, we are interested in whether analogues of these results hold for expansions in non-integer bases. Our first result in the setting of β-expansions is the following. Theorem 2.1. (1) Let β ∈ (1,β ). Then every x ∈ (0, 1 ) has a simply normal KL β−1 β-expansion. √ (2) Let β ∈ (1, 1+ 5). Then every x ∈ (0, 1 ) has a β-expansion for which the fre- 2 β−1 quency of zeros does not exist. √ (3) Let β ∈ (1, 1+ 5). Then there exists c = c(β) > 0 such that for every x ∈ (0, 1 ) 2 β−1 and p ∈ [1/2−c,1/2+c], there exists a β-expansion of x with frequency of zeros equal to p. The quantity β ≈ 1.787 appearing in statement 1 of Theorem 2.1 is the Komornik- KL Loreti constant introduced in [27]. In [27] Komornik and Loreti proved that β is the KL smallest base for which 1 has a unique β-expansion. It has since been shown to be important for many other reasons. We elaborate on the significance of this constant and its relationship with the Thue-Morse sequence in Section 3. Note that we can explicitly calculate a lower bound for the quantity c appearing in statement 3 of Theorem 2.1. We include some explicit calculations in Section 6. It follows from the results listed above that the set of x whose binary expansion is not simply normal has Hausdorff dimension 1. Our next result shows that as β approaches 2 we see a similar phenomenon. Theorem 2.2. (cid:16)(cid:110) (cid:111)(cid:17) limdim x : x has no simply normal β-expansion = 1. H β(cid:37)2 2.2. Hybrid expansions. In this section we consider β-expansions where our digit set is {−1,1} instead of {0,1}. Given β ∈ (1,2) and x ∈ [ −1 , 1 ], we say that a sequence β−1 β−1 ((cid:15) ) ∈ {−1,1}N is a hybrid expansion of x if the following holds: i ∞ (cid:88) (cid:15) i x = βi i=1 and n 1 (cid:88) x = lim (cid:15) . i n→∞ n i=1 Hybrid expansions were first introduced by Gu¨ntu¨rk in [21]. Interestingly, the original motivation for studying hybrid expansions was to overcome the problem of analogue to 4 SIMON BAKER digital conversion where the underling system has background noise. In [21] the following result was asserted without proof. Theorem 2.3. There exists C > 0, such that for all β ∈ (1,1 + C ) there exists c = 1 1 c(β) > 0, such that every x ∈ [−c,c] has a hybrid expansion. A proof was subsequently provided by Dajani, Jiang, and Kempton in [12]. They showed that one can take C ≈ 0.327. We improve upon this theorem in the following way. 1 √ Theorem 2.4. Let β ∈ (1, 1+ 5). Then there exists c = c(β) > 0 such that every x ∈ 2 [−c,c] has a hybrid expansion. It would be desirable to obtain a result of the form: there exists C > 0 such that for every β ∈ (1,1 + C) every x ∈ (− 1 , 1 ) has a hybrid expansion. However, it is an β−1 β−1 immediate consequence of the definition that if x has a hybrid expansion then x ∈ [−1,1]. Since [−1,1] (cid:40) ( −1 , 1 ) for all β ∈ (1,2) it is clear that such a result is not possible. β−1 β−1 Note that if we normalised by a function that decayed at a slower rate than n−1 we would not necessarily have this obstruction. The following result shows that if we replace n−1 with another normalising function that satisfies a certain growth condition, then we have our desired result. √ Theorem 2.5. Let β ∈ (1, 1+ 5). Then there exists c = c(β) > 0 such that if f : N → 2 (0,∞) is a strictly increasing function which satisfies limsupf(n+1)−f(n) < c n→∞ and lim f(n) = ∞, n→∞ then for every x ∈ (− 1 , 1 ) there exists ((cid:15) ) ∈ {−1,1}N such that β−1 β−1 i ∞ (cid:88) (cid:15) i x = βi i=1 and n 1 (cid:88) x = lim (cid:15) . i n→∞ f(n) i=1 The following corollary is an immediate consequence of Theorem 2.5. DIGIT FREQUENCIES AND SELF-AFFINE SETS WITH NON-EMPTY INTERIOR 5 √ Corollary 2.6. Let β ∈ (1, 1+ 5). Then for every x ∈ (− 1 , 1 ) there exists ((cid:15) ) ∈ 2 β−1 β−1 i {−1,1}N such that ∞ (cid:88) (cid:15) i x = βi i=1 and n 1 (cid:88) x = lim (cid:15) . n→∞ n1/2 i i=1 2.3. A family of overlapping self-affine sets and simultaneous expansions. Let {S }m be a collection of contracting maps acting on Rd. A result of Hutchinson [23] j j=1 states that there exists a unique non-empty compact set Λ ⊆ Rd such that m (cid:91) Λ = S (Λ). j j=1 We call Λ the attractor associated to {S }. Often one is interested in determining the j topological properties of Λ. When the collection {S } consists solely of similarities than j the attractor Λ is reasonably well understood. However, when the collection {S } contains j affine maps the situation is known to be much more complicated. In this paper we focus on the following family of self-affine sets. Let 1 < β ,β ,β ≤ 2 1 2 3 and (cid:16)x−1 x−1(cid:17) (cid:16)x+1 x+1(cid:17) S (x,y) = , and S (x,y) = , . −1 1 β β β β 1 2 1 3 For this collection of contractions we denote the associated attractor by Λ . In Figure β1,β2,β3 1 we include some examples. Figure 1. A plot of Λ ,Λ ,Λ 2,1.81,1.66 1.66,1.33,1.53 1.2,1.11,1.05 When β = β we denote Λ by Λ . The case where β = β was studied in 2 3 β1,β2,β3 β1,β2 2 3 [12] and [22]. One problem the authors of these papers were particularly interested in was 6 SIMON BAKER determining those pairs (β ,β ) for which the attractor Λ has non-empty interior. The 1 2 β1,β2 best result in this direction is the following result due to Hare and Sidorov [22]. Theorem 2.7. If β (cid:54)= β and 1 2 (cid:12)β8 −β8(cid:12) (cid:12)β7β7(β −β )(cid:12) (2.1) (cid:12) 2 1(cid:12)+(cid:12) 2 1 2 1 (cid:12) ≤ 2. (cid:12)β7 −β7(cid:12) (cid:12) β7 −β7 (cid:12) 2 1 2 1 Then Λ has non-empty interior and (0,0) ∈ Λo. β1,β2 Let π denote the projection from R2 onto the x-axis. For each x ∈ π(Λ ) let β1,β2,β3 Λx := {y ∈ R : (x,y) ∈ Λ }. β1,β2,β3 β1,β2,β3 We call Λx the fibre of x. Note that π(Λ ) = [ −1 , 1 ]. The following state- β1,β2,β3 β1,β2,β3 β1−1 β1−1 ment is our main result for Λ . β1,β2,β3 √ Theorem 2.8. Let β ∈ (1, 1+ 5). Then there exists c = c(β ) > 0 such that for all 1 2 1 β ,β ∈ (1,1+c) and x ∈ (− 1 , 1 ) the fibre Λx contains an interval. Moreover 2 3 β1−1 β1−1 β1,β2,β3 Λ has non-empty interior and (0,0) ∈ Λo β1,β2,β3 β1,β2,β3 We emphasise that Theorem 2.8 covers the case where β (cid:54)= β . Our approach lends 2 3 itself to explicit calculation and following our method one can obtain a lower bound for the value c appearing in Theorem 2.8. We include some explicit calculations in Section 6. √ Note that for any β sufficiently close to 1+ 5 the set of β ∈ (1,2) satisfying (2.1) 1 2 2 is empty. Consequently Theorem 2.8 provides new examples of β ,β for which Λo is 1 2 √ β1,β2 non-empty. Theorem 2.8 is also optimal in the following sense. For any β ∈ [1+ 5,2) and 1 2 β ,β ∈ (1,2), there exists x ∈ (− 1 , 1 ) such that the fibre Λx is countable and 2 3 β1−1 β1−1 β1,β2,β3 therefore does not contain an interval. We explain why this is the case in Section 7. It is natural to ask whether the property Λx contains an interval for every x ∈ β1,β2,β3 (− 1 , 1 ) is stronger than the property Λo (cid:54)= ∅. This is in fact the case and is a β1−1 β1−1 β1,β2,β3 consequence of the following proposition. Proposition 2.9. Λo (cid:54)= ∅ if and only if {x : Λx contains an interval} contains β1,β2,β3 β1,β2,β3 an open dense subset of [ −1 , 1 ]. β1−1 β1−1 Proof. Letusstartbyintroducingsomenotation. LetF = {x : Λx contains an interval}. β1,β2,β3 Suppose Λo (cid:54)= ∅. Then there exists I and J two nontrivial open intervals such that β1,β2,β3 I×J ⊆ Λ . Let φ (x) = x−1 and φ (x) = x+1. Since S (I×J) is an open rectangle β1,β2,β3 −1 β1 1 β1 −1 contained in Λ , it follows that φ (I) ⊆ F. Similarly φ (I) ⊆ F. Repeating this β1,β2,β3 −1 1 argument, it follows that all images of I under finite concatenations of φ and φ are −1 1 DIGIT FREQUENCIES AND SELF-AFFINE SETS WITH NON-EMPTY INTERIOR 7 contained in F. The union of these images of I is an open dense subset of [ −1 , 1 ]. It β1−1 β1−1 follows that F contains an open dense subset of [ −1 , 1 ]. β1−1 β1−1 It remains to prove the leftwards implication. We start by partitioning the set F. Given (a,b,c,d) ∈ Z4 let (cid:110) (cid:104)a c(cid:105) (cid:111) F := x : , ⊆ Λx . a,b,c,d b d b1,b2,b3 Importantly we have (cid:91) F = F . a,b,c,d (a,b,c,d)∈Z4,a/b 0, such that for 1 < β < β < 1 2 1 + C, there exists c = c(β ,β ) > 0 such that every x ∈ (−c,c) has a simultaneous 1 2 (β ,β ) expansion. Note that the existence of C > 0 satisfying the above follows if one 1 2 can show that for 1 < β < β < 1+C the attractor Λ contains (0,0) in its interior. 1 2 β1,β2 Using this observation Gu¨ntu¨rk’s assertion was proved to be correct in [12]. The largest parameter space for which it is known that (0,0) ∈ Λo , and consequently that any x β1,β2 sufficiently close to zero has a simultaneous (β ,β ) expansion, is that stated in Theorem 1 2 2.7. Ourcontributioninthisdirectionisthefollowingtheoremthatfollowsasanimmediate consequence of Theorem 2.8 by taking β = β . 2 3 √ Theorem 2.10. Let β ∈ (1, 1+ 5). Then there exists C = C(β ) > 0 such that if 1 2 1 β ∈ (1,1+C), then every x sufficiently small has a simultaneous (β ,β )-expansion. 2 1 2 Before moving onto our proofs we say a few words about the methods used in this paper and compare them with those used in [12] and [22]. In these papers the authors show that (0,0) ∈ Λo by constructing a polynomial P(x) = xn+b xn−1+b x+b which satisfies: β1,β2 n−1 1 0 (1) P(β ) = P(β ) = 0 1 2 (2) (cid:80)n−1|b | ≤ 2 j=0 j (3) b = 0 1 (4) b (cid:54)= 0. 0 Once the existence of this polynomial is established, one can devise an algorithm which can be applied to any x ,x sufficiently small, this algorithm then yields an ((cid:15) ) ∈ {−1,1}N 1 2 i such that (x ,x ) = ((cid:80)∞ (cid:15) β−i,(cid:80)∞ (cid:15) β−i). 1 2 i=1 i 1 i=1 i 2 DIGIT FREQUENCIES AND SELF-AFFINE SETS WITH NON-EMPTY INTERIOR 9 This approach is somewhat unsatisfactory. The existence of the polynomial and the algorithm used to construct the ((cid:15) ) provide little intuition as to why (0,0) should be in i the interior of Λ . Our approach, as well as allowing for different rates of contraction β1,β2 in the vertical direction, is more intuitive and explicitly constructs the interval appearing in each fibre of Λ . β1,β2,β3 The rest of this paper is arranged as follows. In Section 3 we recall and prove some technical results that are required to prove our theorems. In Section 4 we prove our theorems relating to digit frequencies. In Section 5 we prove Theorem 2.8. In Section 6 we include an example where we explicitly calculate some of the parameters appearing in our theorems. In Section 7 we include some general discussion and pose some questions. 3. Preliminaries In this section we prove some useful technical results and recall some background mater- ial. Let us start by introducing the maps T (x) = βx+1, T (x) = βx and T (x) = βx−1. −1 0 1 Given an x ∈ I we let β ∞ (cid:110) (cid:88) (cid:15) (cid:111) N i Σ (x) := ((cid:15) ) ∈ {0,1} : = x β i βi i=1 and (cid:110) (cid:111) Ω (x) := (a ) ∈ {T ,T }N : (a ◦···◦a )(x) ∈ I for all n ∈ N . β i 0 1 n 1 β Similarly, given x ∈ I(cid:101) := [ −1 , 1 ] let β β−1 β−1 ∞ (cid:110) (cid:88) (cid:15) (cid:111) N i Σ(cid:101) (x) := ((cid:15) ) ∈ {−1,1} : = x β i βi i=1 and (cid:110) (cid:111) Ω(cid:101) (x) := (a ) ∈ {T ,T }N : (a ◦···◦a )(x) ∈ I(cid:101) for all n ∈ N . β i −1 1 n 1 β The dynamical interpretation of β-expansions is best seen through the following result. Lemma 3.1. For any x ∈ I (x ∈ I(cid:101) ) we have Card Σ (x) = Card Ω (x)(Card Σ(cid:101) (x) = β β β β β Card Ω(cid:101) (x)). Moreover, the map which sends ((cid:15) ) to (T ) is a bijection between Σ (x) and β i (cid:15)i β Ω (x)(Σ(cid:101) (x) and Ω(cid:101) (x)). β β β Lemma 3.1 was originally proved in [6] for an arbitrary digit set of the form {0,...,m}. The proof easily extends to the digit set {−1,1}. Lemma 3.1 allows us to reinterpret problems from β-expansions in terms of the allowable trajectories that can occur within a dynamical system. In Figure 3 we include a graph of T 0 and T acting on I . One can see from this picture, or check by hand, that if x ∈ [1, 1 ] 1 β β β(β−1) 10 SIMON BAKER 0 1 1 1 β−1 β β(β−1) Figure 3. The overlapping graphs of T and T . 0 1 then both T and T map x into I . Therefore, by Lemma 3.1, this x has at least two 0 1 β β-expansions. More generally, if there exists a sequence of T ’s and T ’s that map x into 0 1 [1, 1 ], then x has at least two β-expansions. β β(β−1) The interval [1, 1 ] is clearly important when it comes to studying Σ (x) and Ω (x). β β(β−1) β β In what follows we let (cid:104)1 1 (cid:105) S := , . β β β(β −1) Another particularly useful interval for studying β-expansions is (cid:104) 1 β (cid:105) O := , . β β2 −1 β2 −1 The analogues of S and O for the digit set {−1,1} are β β (cid:104) β −2 2−β (cid:105) (cid:104) 1−β β −1 (cid:105) S(cid:101) := , and O(cid:101) := , . β β(β −1) β(β −1) β β2 −1 β2 −1

Digit frequencies and self-affine sets with non-empty interior

Similar Digit frequencies and self-affine sets with non-empty interior