Exceptional digit frequencies and expansions in non-integer bases

In this paper we study the set of digit frequencies that are realised by elements of the set of β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}-expansions. The main result of this paper demonstrates that as β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} approaches 1, the set of digit frequencies that occur amongst the set of β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}-expansions fills out the simplex. As an application of our main result, we obtain upper bounds for the local dimension of certain biased Bernoulli convolutions.


Introduction
Let M ∈ N and β ∈ (1, M + 1].For any x ∈ I β,M := [0, M β−1 ] there exists a sequence (a i ) ∈ {0, . . ., M} N such that We call such a sequence a β-expansion of x.Note that x has a β-expansion if and only if x ∈ I β,M .Expansions of this type were pioneered in the papers of Parry [19] and Rényi [20].When β = M + 1 then we are in the familiar setting of integer base expansions, where every x ∈ [0, 1] has a unique (M + 1)-expansion, apart from a countable set of points that have precisely two.However, when β ∈ (1, M + 1) the set of β-expansions can exhibit far more exotic behaviour.
This paper is motivated by the following general question.Suppose we are interested in some property of a sequence (a i ) ∈ {0, . . ., M} N .Properties we might be interested in could be combinatorial, number theoretic, or statistical.Can we put conditions on β, such that every x ∈ (0, M β−1 ) admits a sequence (a i ) ∈ Σ β,M (x) that satisfies this property?Alternatively, can we put conditions on β, such that Lebesgue almost every x ∈ (0, M β−1 ) admits a sequence (a i ) ∈ Σ β,M (x) that satisfies this property?Since an x may well have infinitely many β-expansions, answering these questions is non-trivial.The general problem put forward here has been studied previously in different guises by several authors, see [1,5,10,14,15,18].In this paper we are interested in those sequences which exhibit exceptional digit frequencies.What exactly we mean by exceptional will become clear.
The digit frequencies of a representation of a real number is a classical subject going back to the pioneering work of Borel [8], and later Besicovitch [7] and Eggleston [12].Despite being a subject that has its origins in the early 20th century, representations of real numbers and their digit frequencies is still motivating researchers.For some recent contributions in this area see [9,13,16] and the references therein.Most of the existing work in this area was done in a setting where the representation is unique.What distinguishes this work is that we are in a setting where the representations are almost certainly not unique.
Combining the results of [1] and [5] the following theorem is known to hold in the case where M = 1.
In the above the number 1.80194 . . . is the unique root of x 3 − x 2 − 2x + 1 = 0 that lies within the interval (1,2).Note that the intervals appearing in the three statements of Theorem 1.1 are optimal.If β ∈ (1.80194 . . ., 2) then there exists an x ∈ (0, 2 ).One might expect that as β approaches 1, the set of realisable frequencies fills out the relevant simplex.In this paper we show this to be the case.We now express this formally. Let We refer to an element of ∆ M as a frequency vector.Given an ǫ > 0 we define We now introduce an important class of algebraic integers.To each n ∈ N we let β n ∈ (1, 2) be the unique zero of that is contained within the interval (1,2).The fact that β n exists and is unique follows from the observations f n (1) = −1, f ′ n (x) ≥ 1 for all x ≥ 1, and f (2) > 0. In this paper we prove the following result.Theorem 1.2.For any M ∈ N and β ∈ (1, β n ), we have for all x ∈ (0, M β−1 ).Theorem 1.2 demonstrates that as β approaches 1, the set ∆ M,β (x) fills out the simplex ∆ M for any x ∈ (0, M β−1 ).The following corollary of Theorem 1.2 follows immediately.
One might also ask whether one can construct β-expansions for which the digit frequencies do not exist.Theorem 1.1 tells us that when M = 1 and β ∈ (1, 1+ √ 5 2 ) every x ∈ (0, 1 β−1 ) has a β-expansion for which the digit frequencies do not exist.The following theorem shows that the same phenomenon persists for larger alphabets.
Note that in Theorem 1.4 we cannot remove the condition #D ≥ 2, or the condition Removal of either of these conditions forces all of the digit frequencies to exist.Note that in Theorem 1.4 we could simply take D = {0, . . ., M}.As such we have the following corollary.
It is natural to wonder how optimal the constant β n is.With this in mind we introduce the following.Given M ∈ N and n ∈ N let By Theorem 1. Theorem 1.6.For any M ∈ N we have In the above and throughout we make use of the standard big O notation.Theorem 1.6 demonstrates that as n tends to infinity, β n becomes a better approximation to the optimal value β * M,n .It is possible to obtain upper bounds for the quantity β * M,n via existing results in [2], and by carefully examining the proof of Theorem 1.6.Omitting the relevant calculations we include in Figure 1 a table of values detailing some upper bounds for β *

2,n
along with some values for β n .
The following corollary is an immediate consequence of Theorem 1.6.
Corollary 1.7.For any M ∈ N we have It is a surprising consequence of Corollary 1.7 that the leading order term for the rate at which β * M,n accumulates to 1 has no dependence of M. The rest of this paper is structured as follows.In Section 2 we recall some useful dynamical preliminaries and prove some technical results that will be required later.In Section 3 we prove Theorem 1.2 and Theorem 1.4.In Section 4 we prove Theorem 1.6.We conclude in Section 5 where we apply our results to obtain bounds on the local dimension of certain biased Bernoulli convolutions.
Lemma 2.1.For any x ∈ I β,M we have card(Σ β,M (x)) = card(Ω β,M (x)).Moreover, the map sending (a i ) to (T a i ) is a bijection between Σ β,M (x) and Ω β,M (x).Lemma 2.1 allows us to reinterpret Theorem 1.2 and Theorem 1.4 in terms of the existence of an element of Ω β,M (x) exhibiting certain asymptotics.With this in mind we introduce the following notation.Given α = (α i ) ∈ ∪ ∞ j=1 {T 0 , . . ., T M } j we let |α| denote its length, and for any k ∈ {0, . . ., M} we let For notational convenience we let {T 0 , . . ., T M } 0 denote the set consisting of the identity map.Given a finite sequence α ∈ {T 0 , . . ., T M } j , we let α k denote its k-fold concatenation with itself, and let α ∞ denote the infinite concatenation of α with itself.We make use of analogous notational conventions for concatenations of finite sequences of digits.
It is straightforward to show that the unique fixed point of T k is k β−1 .It is also straightforward to show that the following equality holds for any x ∈ R and l ∈ N (2.1) Despite being a simple observation, equation (2.1) will be extremely useful.We point out here three immediate consequences of it without proof.
(1) Let ǫ > 0. There exists L ∈ N depending only upon M, β and ǫ, such that for any (2) Let ǫ > 0. There exists L ∈ N depending only upon M, β and ǫ, such that for any for some 1 ≤ l ≤ L. (3) Let ǫ > 0. There exists L ∈ N depending only upon M, β and ǫ such that if Statement (3) in Lemma 2.2 follows from the previous two statements and the observations In our applications the ǫ appearing in Lemma 2.2 will often be a function of β.Therefore the relevant L will often only depend upon M and β.Despite being a fairly trivial observation, Lemma 2.2 will be useful throughout.In particular statement (3).
When constructing β-expansions that satisfy certain asymptotics, it is useful to partition the interval I β,M into subintervals for which we have a lot of control over how the different T k behave.This technique was originally used in [2] and [4] to study the size of Σ β,M (x).
In the literature S β,k 1 ,k 2 is commonly referred to as the switch region corresponding to k 1 and k 2 .
We refer the reader to Figure 2 for a diagram detailing the above.
The following lemma proves that a useful class of subintervals depending on n, k 1 , and k 2 , will always be contained in the switch region corresponding to k 1 and k and .
Proof.Fix n ∈ N. We will only show that (2.2) holds.The proof that (2.3) holds follows similarly.
It suffices to show that for any β ∈ (1, Rewriting the right hand side of (2.4) using the geometric series formula we see that (2.4) is equivalent to Comparing coefficients, we see that this inequality is equivalent to Cancelling coefficients and using the geometric series formula, we can rewrite this expression as ) is satisfied, and consequently (2.4) is also satisfied.We now turn our attention to (2.5).Expressing the left hand side of (2.5) using the geometric series formula, we see that it is equivalent to Comparing coefficients, we see that this is equivalent to Cancelling coefficients and using the geometric series formula, we can then rewrite this expression as When n = 1 we can verify that (2.7) holds whenever

This is true for any
2 ).Therefore (2.5) holds for β ∈ (1, 2 ).To complete the proof of our lemma one has to verify that for any n ≥ 1.However, this is a straightforward consequence of the definition of In what follows it is useful to define the following intervals, let ,n so our notation may at first seem redundant.However it will be useful to adopt both conventions.As we will see, within the interval D β,M,n we shall apply many different T k 's to construct a β-expansion with certain asymptotics.Importantly these T k 's will never map the point in question out of D β,M,n , so this interval can be thought of as the state space of our construction.When we want to emphasise the state space interpretation we will use the notation D β,M,n .Within an interval D β,k 1 ,k 2 ,n we will be able to build an element of ∪ ∞ j=1 {T k 1 , T k 2 } j which exhibits certain asymptotics.When we want to emphasise the fact we are building a sequence consisting of T 0 's and T M 's with certain asymptotics we will use the notation D β,0,M,n .
The following lemma establishes the existence of an upper bound for the number of maps required to map a point from the interior of I β,M into the interior of an , and δ > 0. There exists L ∈ N and ǫ > 0, such that for any Proof.We show that for each We now proceed via a case analysis.
] automatically by our hypothesis.So we can take L 0,M = 0.
It is a straightforward calculation to prove that for any β ∈ (1, 2) and k 1 < k 2 we have It therefore suffices to show that there exists The existence of the desired ǫ k 1 ,k 2 will then follow by (2.8). Let then there is nothing to prove.Suppose this is not the case and Consequently, it suffices to consider the case where  (2.8), and (2.9), when combined then imply that there exists L 3 ∈ N, such that for some l 3 ≤ L 3 we have We observe from equation (2.10) that x has been mapped from we can repeat the steps used to derive (2.10) from (2.9) and map the orbit of x into the interval [Π β ((j + 1, j + 4) ∞ ), Π β ((j + 4, j + 1) ∞ )] using only a bounded number of maps.Repeatedly applying this procedure we see that the orbit of x must eventually be mapped into ] by some η as required.Moreover, since the number of maps needed to map a point from ] can always be bounded above by some constant, it follows that length of η can always be bounded above by some We can bound the number of maps required to do this by Lemma 2.2.
Case 3. (k 1 > 0, k 2 = M).For any β > 1 we always have the inclusion Therefore it suffices to show that there exists then by Lemma 2.2 it follows that there exists ) then one can replicate the argument used in Case 2 to deduce that x can be mapped using only a bounded number of maps into the interval for some ǫ k 1 −1,k 1 > 0. Consequently, without loss of generality we may assume that At this point we make a simple observation.For any β ∈ (1, 2) and 0 < k 1 < M we have Therefore, by continuity there exists parameters γ > 0 and κ > 0 such that if , then x is a uniformly bounded distance away from the fixed point k 1 β−1 .Therefore, by Lemma 2.2 our point x can be mapped into is a uniformly bounded distance away from the endpoints of I β,k 1 ,M .Therefore by Lemma 2.2 we can bound the number of maps required to map In the first case we proceed as in the case where l 2 = 0 and apply In the second case we are already a uniformly bounded distance away from the endpoints of ] using a bounded number of maps.The following lemma tells us that if x is in the interior of an I β,k 1 ,k 2 and a bounded distance from its endpoints, then we can map x into the intervals appearing in Lemma 2.3 using a bounded number of maps.Lemma 2.5.Let M ∈ N, β ∈ (1, β n ) and ǫ > 0. There exists L ∈ N, such that for any Similarly, there exists Proof.Let us start by fixing We will only show that there exists ) holds follows by a similar argument.To finish the proof of the lemma we take there is nothing to prove.Suppose The fact the number of iterations of T k 2 required to do this is bounded also follows from Lemma 2.2 and the fact x < k .
] is some uniformly bounded distance away from k 2 β−1 .We now repeat our initial argument in the case where The following lemma follows from the proof of Lemma 2.5.It poses greater restrictions on the orbit of x under η.Lemma 2.6.Let M ∈ N and β ∈ (1, β n ).Then there exists L ∈ N, such that for any Lemma 2.6 gives conditions ensuring that the orbit of x under η stays within the interval D β,k 1 ,k 2 ,n .This property will be useful when we want our orbit to be mapped into yet another The following lemma shows that if x is contained in D β,M,n , then x can be mapped into the intervals appearing in Lemma 2.3 using a bounded number of maps.Lemma 2.7.Let M ∈ N and β ∈ (1, β n ).Then there exists L ∈ N, such that for any x ∈ D β,M,n and k 1 , k 2 ∈ {0, . . ., M} satisfying k 1 < k 2 , there exists Similarly, there exists Proof.Lemma 2.7 follows almost immediately from Lemma 2.4 and Lemma 2.5.We include the proof for completion.Let us start by emphasising D β,M,n ⊆ (0, M β−1 ) for β ∈ (1, β n ) and so is contained in [δ, M β−1 − δ] for some δ depending on M and β.Now fix x ∈ D β,M,n .By Lemma 2.4 there exists a bounded number of transformations that map x into [ allows us to assert that there exists a bounded number of maps that map this image of Hence our η 1 exists.The existence of η 2 follows from an analogous argument.

Proofs of Theorems 1.2 and 1.4
We now proceed with our proof of Theorem 1.2.Our proof relies on the following two propositions.
Proposition 3.1.Let M ∈ N and β ∈ (1, β n ).There exists C > 0 such that, for any Proof.The proof of Proposition 3.1 relies on devising an algorithm that yields the desired τ .At each step in the algorithm we should check rigorously that properties (1), (2), and (3) hold.However, for the sake of brevity we simply state here that property (1) will hold since τ will be constructed by concatenating maps of the form guaranteed by Lemma 2.6, maps from Also, note that since τ will be an element of {T k 1 , T k 2 } N , property ( 2) is equivalent to property (3).So it suffices to prove property (2).
By Lemma 2.6 we can assume without loss of generality that Step 1.
for some n ≤ l 1 ≤ n ′ by Lemma 2.2 and Lemma 2.3.Importantly n ′ depends solely upon M and β.Let τ 1 = (T k 2 , T l 1 k 1 ).Note that Step 2. At this point we remark that This is because p k 1 ≤ n n+1 and l 1 ≥ n.We now apply Lemma 2.6 to τ 1 (x).There exists We let κ 1 = (τ 1 , η 1 ) and observe At this point we examine the sign of ) is negative we stop and let τ 2 = κ 1 .Note that if this is the case then by (3.1) we have Suppose (3.4) is positive, we then use (3.2) to assert that there exists n ≤ l 2 ≤ n ′ such that Where n ′ is as above and depends only upon M and β.Letting We trivially have Equation (3.6) when combined with (3.3) and the assumption (3.4) is positive implies At this point we examine the sign of If (3.8) is negative we stop and let τ 2 = κ 2 .In which case If (3.8) is positive then we repeat our previous step with κ 1 (x) replaced with κ 2 (x).So there exists n ≤ l 3 ≤ n ′ such that Then by (3.5) and (3.7) we have Moreover, we also have | is positive we can repeat this process.We iteratively define κ 4 , κ 5 , . . ., etc.We stop if for some j.Assume j * is the smallest such j such that (3.11) occurs.We then let τ 2 = κ j * .Importantly, analogues of (3.9) and (3.10) hold for each intermediate κ term.Consequently We also remark that at this point If there does not exist a j such that (3.11) occurs, then we let τ ∈ {T k 1 , T k 2 } N be the infinite sequence we attain as the limit of the κ j .Since each κ j is a prefix of κ j ′ for any j ′ > j the infinite sequence τ is well defined.In this case the following holds for each j ∈ N Consequently, So in this case τ would satisfy property (2).
Step j + 1. Suppose we have constructed τ j such that Moreover, assume τ j satisfies (3.12) Note that at the end of step 2 the sequence τ 2 we've constructed satisfies the first condition and (3.13).Assume (3.13) holds, the case where (3.12) holds is handled similarly.We now essentially repeat the argument given in Step 2. By Lemma 2.6 we can map We stop if we observe a change of sign.If we observe a change of sign the sequence we will have constructed is our τ j+1 .It can be shown that this τ j+1 will then satisfy as required.If we never observe a sign change, then the infinite sequence we attain is our desired τ .It satisfies Clearly we can either repeat step j + 1 indefinitely, in which case the infinite limit of the τ j 's will satisfy property (2) by (3.14), or at some point τ j does not give rise to an τ j+1 .This is the case where we do not observe a sign change.In this case, the infinite sequence we would obtain by repeatedly applying either . In either case the desired τ exists and we can take C = 2n ′ + L + 2. Proposition 3.1 has the useful consequence that for any N ∈ N, the sequence (τ 1 , . . ., τ N ) has k 1 frequency approximately p k 1 and k 2 frequency approximately p k 2 .We will use this fact in the proof of Theorem 1.2.
Given ǫ > 0 recall that Clearly ∆ M,ǫ is a compact, convex subset of R M +1 .It is a simple exercise to check that it's extremal points are all the vectors of the form q ǫ,k 1 ,k 2 = (q ǫ,0 , . . ., q ǫ,M ) where all entries are zero apart from q ǫ,k where all entries are zero apart from v k 1 = n n+1 and v k 2 = 1 n+1 .In the following we denote the convex hull of a finite set of vectors by Conv(•).
Proof.By the Krein-Milman theorem (see [11]) it suffices to check that the extremal points of ∆ M,ǫ are in the convex hull of Equipped with Proposition 3.1 and Proposition 3.2 we are now in a position to prove Theorem 1.2.Before giving our proof we give an outline of our argument.Suppose β ∈ (1, β n ) and p ∈ ∆ M, 1

n+1
. By Proposition 3.1 we know that for each k 1 = k 2 we can construct finite sequences of maps τ k 1 ,k 2 of an arbitrary length with frequencies equal to n n+1 and Evaluating (3.15) and using the definition of v n,k 1 ,k 2 we have the following componentwise formula, (3.16) for any k ∈ {0, . . ., M}. Labelling terms we can write {(k 1 , k 2 ) : . Let us now fix a sequence of natural numbers (N i ) such that N i → ∞, (3.17For example one could take N i = i 2 .To each (k p 1 , k p 2 ) and i ∈ N we associate Where ⌊•⌋ denotes the integer part.We now devise an algorithm to construct the desired α.
Step 1.Without loss of generality we may assume that 2 ,n we can apply Lemma 2.7 to τ 1,1 (x) to obtain η 1,1 such that |η 1,1 | ≤ L and ,n , we know by Proposition 3.1 that there exists τ 1,2 such that (τ 2 ,n we can apply Lemma 2.7 to (τ We repeat this argument until eventually we obtain a sequence ψ 1 of the form for all 1 ≤ p ≤ M(M + 1).Let α 1 = ψ 1 .
By property (1) above it follows that α j is prefix of α j ′ for any j ′ > j, so the limiting infinite sequence α is well defined.Clearly α ∈ Ω β,M (x) by property (1).By Lemma 2.1 to prove our theorem it remains to show that α satisfies the required digit frequency properties.
Observe that for any i ∈ N we have Using (3.19) we obtain Applying (3.16) we see that For any n ∈ N there exists j ∈ N such that |α j | ≤ n < |α j+1 |.Therefore for any k ∈ {0, . . ., M} we have By (3.24).
It follows from the above, (3.17), and (3.18) that lim sup Since k was arbitrary this completes our proof.
Proof of Theorem 1.4.The proof of Theorem 1.4 is an adaptation of the proof of Theorem 1.2.As such we just provide an outline and leave the details to the interested reader.Let D and (p k ) k∈D c be as in the statement of Theorem 1.4.Under the assumptions of this theorem there exists q ∈ ∆ M, 1 n+1 and q ′ ∈ ∆ M, 1 n+1 such that q k = q ′ k for all k ∈ D, and we can carefully go through the argument given in the proof of Theorem 1.2 to construct an α ∈ Ω β,M (x) satisfying the following three properties: (1) There exists a sequence (N p ) such that (2) There exists a sequence (N j ) such that Since q k = q ′ k for all k ∈ D it follows that α satisfies the required digit frequency properties.To construct the α ∈ Ω β,M (x) described above one proceeds initially as in the proof of Theorem 1.2 as if we were trying to build an expansion with digit frequencies described by the vector q.Once we have a sufficiently good approximation to q we change our algorithm to construct an expansion with digit frequencies described by q ′ , then once we have a sufficiently good approximation to q ′ we switch back to q and so on.

Proof of Theorem 1.6
In this section we prove Theorem 1.6.We split the proof into the following two propositions.Proof.Recall that β n is the unique solution in (1, 2) of the polynomial Evaluating f n at 1 + log n−log log n n we obtain To obtain the third line above we used the fact that (1 + x n ) n ≤ e x for all x ≥ 0. Since f n (1) < 0 and f ′ n (x) > 1 for all x ≥ 1 it follows from the above that 1+ We will show that has Lebesgue measure zero for all β satisfying (4.2).We start by remarking that (4.3) for any k ∈ {0, . . ., M} and ǫ > 0. The right hand side of (4.3) is a limsup set, so by the Borel Cantelli lemma to prove that the set on the right hand side of has zero Lebesgue measure, and therefore the left hand side has zero Lebesgue measure, it suffices to show that Where here and throughout L denotes the Lebesgue measure.This bound follows immediately since Where the final inequality holds by (4.Combining Proposition 4.1 and Proposition 4.2 we deduce Theorem 1.6.

Applications to biased Bernoulli convolutions
Given M ∈ N and q = (q 0 , . . ., q M ) ∈ ∆ M , one can define a product measure P q := ∞ 1 {q 0 , . . ., q M } on {0, . . ., M} N .Given a β > 1, one then defines the Bernoulli convolution associated to q via the equation µ q (E) = P q (a i ) ∈ {0, . . ., M} N : where E is an arbitrary Borel subset of R. Bernoulli convolutions have been studied since the 1930's.They've connections with algebraic numbers, dynamical systems, and fractal geometry.The fundamental question surrounding Bernoulli convolutions is to determine those β ∈ (1, M + 1] and q ∈ ∆ M such that µ q is absolutely continuous with respect to the Lebesgue measure.For more on this class of measures we refer the reader to the surveys of Peres, Schlag, and Solomyak [24], and Varju [25]. Corollary 5.2.Let M ∈ N and β ∈ (1, β n ).Then for any q as above and x ∈ (0, M β−1 ) we have d µq (x) ≤ − (n log q * + log q * * ) (n + 1) log β .

Case 4 .
(k 1 = 0, k 2 < M).The proof of Case 4 is analogous to the the proof of Case 3.

Proposition 4 . 1 .
For all n ∈ N we have1 + log n − log log n n < β n .
1 n+1 up to a bounded error term.By Proposition 3.2 we know that p∈ Conv({v n,k 1 ,k 2 } k 1 ,k 2 ).This proposition guarantees the existence of weights that may be used to construct p from the frequencies of the different τ k 1 ,k 2 .The problem is that we cannot freely concatenate the τ k 1 ,k 2 .We have to travel between the D β,k 1 ,k 2 ,n which introduces an error.However, by Lemma 2.7 this error can always be bounded.Consequently by taking repeatedly larger τ k 1 ,k 2 this error becomes progressively more negligible, meaning the limiting sequence we construct will achieve the desired frequency p. ) we have p ∈ ∆ M,β (x).By Proposition 3.2 there exists r k 1 ,k 2 for all k 1 ).What is more, for any i ∈ N and k ∈ {0, ..., M}, it follows from properties (2) and (4), and equations (3.20) and (3.21) that i | − N i ≤ (L + 1)M(M + 1)for all i ∈ N.