# Exceptional digit frequencies and expansions in non-integer bases

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## Abstract

In this paper we study the set of digit frequencies that are realised by elements of the set of \(\beta \)-expansions. The main result of this paper demonstrates that as \(\beta \) approaches 1, the set of digit frequencies that occur amongst the set of \(\beta \)-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.

## Keywords

Expansions in non-integer bases Digit frequencies Bernoulli convolutions## Mathematics Subject Classification

Primary 11A63 Secondary 11K16 11K55## 1 Introduction

*x*. Note that

*x*has a \(\beta \)-expansion if and only if \(x\in I_{\beta ,M}.\) Expansions of this type were pioneered in the papers of Parry [19] and Rényi [20]. When \(\beta =M+1\) then we are in the familiar setting of integer base expansions, where every \(x\in [0,1]\) has a unique \((M+1)\)-expansion, apart from a countable set of points that have precisely two. However, when \(\beta \in (1,M+1)\) the set of \(\beta \)-expansions can exhibit far more exotic behaviour.

*x*is an endpoint of \(I_{\beta ,M}\) then \(\Sigma _{\beta ,M}(x)\) is always either \(\{(0)^{\infty }\}\) or \(\{(M)^{\infty }\}\). As such, all of the interesting behaviour occurs within the interior of \(I_{\beta ,M}.\) For \(\beta \in [\mathcal {G}(M),M+1)\) the cardinality of \(\Sigma _{\beta ,M}(x)\) for a generic

*x*is best described by a result of Sidorov, see [21, 22]. This result implies that for any \(\beta \in [\mathcal {G}(M),M+1),\) we have \(\text {card}(\Sigma _{\beta ,M}(x))=2^{\aleph _0}\) for Lebesgue almost every \(x\in I_{\beta ,M}\). We remark that all other possible values of \(\text {card}(\Sigma _{\beta ,M}(x))\) are achievable. That is, for any \(k\in \mathbb {N}\cup \{\aleph _0\},\) there exists \(\beta \in (1,2)\) and \(x\in (0,\frac{1}{\beta -1})\) such that \(\text {card}(\Sigma _{\beta ,1}(x)) =k,\) see [3, 6, 23] and the references therein.

This paper is motivated by the following general question. Suppose we are interested in some property of a sequence \((a_i)\in \{0,\ldots ,M\}^{\mathbb {N}}\). Properties we might be interested in could be combinatorial, number theoretic, or statistical. Can we put conditions on \(\beta ,\) such that every \(x\in (0,\frac{M}{\beta -1})\) admits a sequence \((a_i)\in \Sigma _{\beta ,M}(x)\) that satisfies this property? Alternatively, can we put conditions on \(\beta ,\) such that Lebesgue almost every \(x\in (0,\frac{M}{\beta -1})\) admits a sequence \((a_i)\in \Sigma _{\beta ,M}(x)\) that satisfies this property? Since an *x* may well have infinitely many \(\beta \)-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.

### 1.1 Statement of results

*k*-frequency of \((a_i)\) to be

*k*-frequency exists and \(\text {freq}_{k}(a_i)=\frac{1}{M+1}\). Borel’s normal number theorem tells us that Lebesgue almost every

*x*has a simply normal \((M+1)\)-expansion, see [8].

Combining the results of [1, 5] the following theorem is known to hold in the case where \(M=1\).

### Theorem 1.1

- 1.
Let \(\beta \in (1,1.80194\ldots ]\). Then for every \(x\in (0,\frac{1}{\beta -1})\) there exists \((a_i)\in \Sigma _{\beta ,1}(x)\) such that \((a_i)\) is simply normal.

- 2.
Let \(\beta \in (1,\frac{1+\sqrt{5}}{2})\). Then for every \(x\in (0,\frac{1}{\beta -1})\) there exists \((a_i)\in \Sigma _{\beta ,1}(x)\) such that the 0-frequency of \((a_i)\) and the 1-frequency of \((a_i)\) both don’t exist.

- 3.
Let \(\beta \in (1,\frac{1+\sqrt{5}}{2})\). Then there exists \(c=c(\beta )>0\) such that for every \(x\in (0,\frac{1}{\beta -1})\) and \(p\in [1/2-c,1/2+c],\) there exists \((a_i)\in \Sigma _{\beta ,1}(x)\) such that \(\text {freq}_{0}(a_i)=p\) and \(\text {freq}_{1}(a_i)=1-p.\)

In the above the number \(1.80194\ldots \) 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 \(\beta \in (1.80194\ldots ,2)\) then there exists an \(x\in (0,\frac{1}{\beta -1})\) with no simply normal \(\beta \)-expansions. Similar statements hold for parts (2) and (3) of Theorem 1.1.

Theorem 1.1 provides no information as to what frequencies are realised as \(\beta \) approaches 1, or what happens for larger alphabets. Examining the techniques used in the proof of statement (3) from Theorem 1.1, we see that they cannot improve upon the estimate \(0<c(\beta )\le 1/6\) for all \(\beta \in (1,\frac{1+\sqrt{5}}{2})\). One might expect that as \(\beta \) 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.

In this paper we prove the following result.

### Theorem 1.2

Theorem 1.2 demonstrates that as \(\beta \) approaches 1, the set \(\Delta _{M,\beta }(x)\) fills out the simplex \(\Delta _{M}\) for any \(x\in (0,\frac{M}{\beta -1})\). The following corollary of Theorem 1.2 follows immediately.

### Corollary 1.3

Let \(M\in \mathbb {N}\) and \(\beta \in (1,\frac{1+\sqrt{5}}{2}).\) Then every \(x\in (0,\frac{M}{\beta -1})\) admits a simply normal \(\beta \)-expansion.

One might also ask whether one can construct \(\beta \)-expansions for which the digit frequencies do not exist. Theorem 1.1 tells us that when \(M=1\) and \(\beta \in (1,\frac{1+\sqrt{5}}{2})\) every \(x\in (0,\frac{1}{\beta -1})\) has a \(\beta \)-expansion for which the digit frequencies do not exist. The following theorem shows that the same phenomenon persists for larger alphabets.

### Theorem 1.4

Note that in Theorem 1.4 we cannot remove the condition \(\#D\ge 2\), or the condition \(\sum _{k\in D^c}p_k<1\). Removal of either of these conditions forces all of the digit frequencies to exist. Note that in Theorem 1.4 we could simply take \(D=\{0,\ldots ,M\}.\) As such we have the following corollary.

### Corollary 1.5

Let \(M\ge 2\) and \(\beta \in (1,\frac{1+\sqrt{5}}{2}).\) Then every \(x\in (0,\frac{M}{\beta -1})\) admits a \(\beta \)-expansion such that \(\text {freq}_k(a_i)\) does not exist for all \(0\le k\le M\).

### Theorem 1.6

In the above and throughout we make use of the standard big \(\mathcal {O}\) notation. Theorem 1.6 demonstrates that as *n* tends to infinity, \(\beta _n\) becomes a better approximation to the optimal value \(\beta _{M,n}^*\). It is possible to obtain upper bounds for the quantity \(\beta _{M,n}^*\) via existing results in [2], and by carefully examining the proof of Theorem 1.6. Omitting the relevant calculations we include in Table 1 a table of values detailing some upper bounds for \(\beta _{2,n}^*\) along with some values for \(\beta _n\).

The following corollary is an immediate consequence of Theorem 1.6.

### Corollary 1.7

*M*.

A table of values for \(\beta _n\)

n | \(\beta _n\) | Upper bound for \(\beta _{2,n}^*\) |
---|---|---|

1 | \(\frac{1+\sqrt{5}}{2}= 1.618\ldots \) | 2 |

2 | \(1.466\ldots \) | 2 |

3 | \(1.380\ldots \) | 2 |

4 | \(1.325\ldots \) | \(1.894\ldots \) |

5 | \(1.285\ldots \) | \(1.761\ldots \) |

10 | \(1.184\ldots \) | \(1.432\ldots \) |

25 | \(1.098\ldots \) | \(1.207\ldots \) |

50 | \(1.058\ldots \) | \(1.116\ldots \) |

100 | \(1.034\ldots \) | \(1.064\ldots \) |

The rest of this paper is structured as follows. In Sect. 2 we recall some useful dynamical preliminaries and prove some technical results that will be required later. In Sect. 3 we prove Theorems 1.2 and 1.4. In Sect. 4 we prove Theorem 1.6. We conclude in Sect. 5 where we apply our results to obtain bounds on the local dimension of certain biased Bernoulli convolutions.

## 2 Preliminaries

### Lemma 2.1

For any \(x\in I_{\beta ,M}\) we have \(\mathrm {card}(\Sigma _{\beta ,M}(x))=\mathrm {card}(\Omega _{\beta ,M}(x)).\) Moreover, the map sending \((a_i)\) to \((T_{a_i})\) is a bijection between \(\Sigma _{\beta ,M}(x)\) and \(\Omega _{\beta ,M}(x).\)

*k*-fold concatenation with itself, and let \(\alpha ^{\infty }\) denote the infinite concatenation of \(\alpha \) with itself. We make use of analogous notational conventions for concatenations of finite sequences of digits.

### Lemma 2.2

- 1.Let \(\epsilon >0.\) There exists \(L\in \mathbb {N}\) depending only upon
*M*, \(\beta \) and \(\epsilon ,\) such that for any \(x>\frac{k}{\beta -1}+\epsilon \) and \(y\in [x,\frac{M}{\beta -1}]\) we havefor some \(1\le l\le L\).$$\begin{aligned} T_{k}^{l}(x)\in [y,T_{k}(y)] \end{aligned}$$ - 2.Let \(\epsilon >0.\) There exists \(L\in \mathbb {N}\) depending only upon
*M*, \(\beta \) and \(\epsilon ,\) such that for any \(x<\frac{k}{\beta -1}-\epsilon \) and \(y\in [0, x]\) we havefor some \(1\le l\le L\).$$\begin{aligned} T_{k}^{l}(x)\in [T_{k}(y),y] \end{aligned}$$ - 3.Let \(\epsilon >0\). There exists \(L\in \mathbb {N}\) depending only upon
*M*, \(\beta \) and \(\epsilon \) such that if \(k_1,k_2\in \{0,\ldots ,M\}\) satisfy \(k_1<k_2\) and \(x\in [\frac{k_1}{\beta -1}+\epsilon ,\Pi _{\beta }((k_1,k_2)^{\infty })],\) thenfor some \(1\le l\le L.\) Similarly, if \(x\in [\Pi _{\beta }((k_2,k_1)^{\infty }),\frac{k_2}{\beta -1}-\epsilon ]\) then$$\begin{aligned} T_{k_1}^l(x)\in [\Pi _{\beta }((k_1,k_2)^{\infty }),\Pi _{\beta }((k_2,k_1)^{\infty })] \end{aligned}$$for some \(1\le l\le L.\)$$\begin{aligned} T_{k_2}^l(x)\in [\Pi _{\beta }((k_1,k_2)^{\infty }),\Pi _{\beta }((k_2,k_1)^{\infty })] \end{aligned}$$

*L*will often only depend upon

*M*and \(\beta \). Despite being a fairly trivial observation, Lemma 2.2 will be useful throughout. In particular statement (3).

When constructing \(\beta \)-expansions that satisfy certain asymptotics, it is useful to partition the interval \(I_{\beta ,M}\) into subintervals for which we have a lot of control over how the different \(T_{k}\) behave. This technique was originally used in [2, 4] to study the size of \(\Sigma _{\beta ,M}(x)\).

*x*such that \(T_{k_1}(x)\in I_{\beta ,k_1,k_2}\) and \(T_{k_2}(x)\in I_{\beta ,k_1,k_2}.\) In the literature \(S_{\beta ,k_1,k_2}\) is commonly referred to as the switch region corresponding to \(k_1\) and \(k_2\). We refer the reader to Fig. 1 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 \(k_2\) for \(\beta \in (1,\beta _n).\)

### Lemma 2.3

### Proof

Fix \(n\in \mathbb {N}\). We will only show that (2.2) holds. The proof that (2.3) holds follows similarly.

The following lemma establishes the existence of an upper bound for the number of maps required to map a point from the interior of \(I_{\beta ,M}\) into the interior of an \(I_{\beta ,k_1,k_2}.\)

### Lemma 2.4

### Proof

**Case 1**\((k_1=0,\, k_2=M)\). If \(k_1=0\) and \(k_2=M\) then let \(\epsilon _{0,M}=\delta .\) We then have \(x\in [\epsilon _{0,M},\frac{M}{\beta -1}-\epsilon _{0,M}]\) automatically by our hypothesis. So we can take \(L_{0,M}=0\).

**Case 2**\((k_1>0,\, k_2<M)\). It is a straightforward calculation to prove that for any \(\beta \in (1,2)\) and \(k_1<k_2\) we have

*x*has been mapped from \(I_{\beta ,j,j+1}\) into \([\Pi _{\beta }((j,j+3)^{\infty }),\Pi _{\beta }((j+3,j)^{\infty })]\). If \(j=k_1-1\) then our proof is complete. If \(j<k_1-1,\) then we can repeat the steps used to derive (2.10) from (2.9) and map the orbit of

*x*into the interval \([\Pi _{\beta }((j+1,j+4)^{\infty }),\Pi _{\beta }((j+4,j+1)^{\infty })]\) using only a bounded number of maps. Repeatedly applying this procedure we see that the orbit of

*x*must eventually be mapped into \([ \Pi _{\beta }((k_1-1,k_2+1)^{\infty }),\Pi _{\beta }((k_2+1,k_1-1)^{\infty })]\) by some \(\eta \) as required. Moreover, since the number of maps needed to map a point from \([\Pi _{\beta }((j-1,j+2)^{\infty }),\Pi _{\beta }((j+2,j-1)^{\infty })]\) into \([\Pi _{\beta }((j,j+3)^{\infty }),\Pi _{\beta }((j+3,j)^{\infty })]\) can always be bounded above by some constant, it follows that length of \(\eta \) can always be bounded above by some \(L_{k_1,k_2}\). If

*x*is mapped into \([ \Pi _{\beta }((k_1-1,k_2+1)^{\infty }),\Pi _{\beta }((k_2+1,k_1-1)^{\infty })]\) by repeatedly applying \(T_{k_1-1}.\) 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 \(\beta >1\) we always have the inclusion

*x*can be mapped using only a bounded number of maps into the interval

If \(l_2=0\) and \(x\in [\frac{k_1}{\beta -1}+\gamma ,\Pi _{\beta }((k_1,M)^{\infty })],\) then *x* is a uniformly bounded distance away from the fixed point \(\frac{k_1}{\beta -1}\). Therefore, by Lemma 2.2 our point *x* can be mapped into \([\Pi _{\beta }((k_1,M)^{\infty }),\Pi _{\beta }((M,k_1-1)^{\infty })]\) by a bounded number of maps. If \(x\in [\frac{k_1}{\beta -1},\frac{k_1}{\beta -1}+\gamma ]\) then we apply \(T_{k_1-1}\) to *x*. By (2.11) the point \(T_{k_1-1}(x)\) is a uniformly bounded distance away from the endpoints of \(I_{\beta ,k_1,M}.\) Therefore by Lemma 2.2 we can bound the number of maps required to map \(T_{k_1-1}(x)\) into \([\Pi _{\beta }((k_1,M)^{\infty }),\Pi _{\beta }((M,k_1)^{\infty })].\)

**Case 4**\((k_1=0,k_2<M)\). The proof of Case 4 is analogous to the the proof of Case 3. \(\square \)

The following lemma tells us that if *x* is in the interior of an \(I_{\beta ,k_1,k_2}\) 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

### Proof

Let us start by fixing \(k_1< k_2\) and \(x\in [\frac{k_1}{\beta -1}+\epsilon ,\frac{k_2}{\beta -1}-\epsilon ].\) We will only show that there exists \(L_{k_1,k_2}\in \mathbb {N}\) and \(\eta ^1\in \cup _{j=0}^{L_{k_1,k_2}}\{T_{k_1},T_{k_2}\}^{j}\) such that (2.12) holds. The existence of an \(L_{k_1,k_2}\) and an \(\eta ^2\in \cup _{j=0}^{L}\{T_{k_1}, T_{k_2}\}^{L}\) such that (2.13) holds follows by a similar argument. To finish the proof of the lemma we take \(L=\max _{k_1,k_2} L_{k_1,k_2}.\)

*x*into \([\Pi _{\beta }((k_1,k_2^n)^{\infty }),\Pi _{\beta }((k_2,k_1,k_2^{n-1})^{\infty })]\) by Lemma 2.2. 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 \(x<\frac{k_2}{\beta -1}-\epsilon \). If \(x<\Pi _{\beta }((k_1,k_2^n)^{\infty })\) then repeated iteration of \(T_{k_1}\) maps

*x*into \([\Pi _{\beta }((k_1,k_2^n)^{\infty }),T_{k_1}(\Pi _{\beta }((k_1,k_2^n)^{\infty }))].\) Lemma 2.2 and the fact \(x>\frac{k_1}{\beta -1}+\epsilon \) implies that the number of maps required to map

*x*into \([\Pi _{\beta }((k_1,k_2^n)^{\infty }),T_{k_1}(\Pi _{\beta }((k_1,k_2^n)^{\infty }))]\) can be bounded above. If

*x*has been mapped into \([\Pi _{\beta }((k_1,k_2^n)^{\infty }),\Pi _{\beta }((k_2,k_1,k_2^{n-1})^{\infty })]\) then we are done. If not then

*x*has been mapped into \((\Pi _{\beta }((k_2,k_1,k_2^{n-1})^{\infty }),T_{k_1}(\Pi _{\beta }((k_1,k_2^n)^{\infty }))].\) Since \(\beta \in (1,\beta _n)\) we know by Lemma 2.3 that

*x*within \((\Pi _{\beta }((k_2,k_1,k_2^{n-1})^{\infty }),T_{k_1}(\Pi _{\beta }((k_1,k_2^n)^{\infty }))]\) is some uniformly bounded distance away from \(\frac{k_2}{\beta -1}.\) We now repeat our initial argument in the case where \(x>\Pi _{\beta }((k_2,k_1,k_2^{n-1})^{\infty })\) to complete our proof. \(\square \)

The following lemma follows from the proof of Lemma 2.5. It poses greater restrictions on the orbit of *x* under \(\eta \).

### Lemma 2.6

Lemma 2.6 gives conditions ensuring that the orbit of *x* under \(\eta \) stays within the interval \(D_{\beta ,k_1,k_2,n}\). This property will be useful when we want our orbit to be mapped into yet another \(D_{\beta ,k_1',k_2',n}.\)

The following lemma shows that if *x* is contained in \(D_{\beta ,M,n},\) then *x* can be mapped into the intervals appearing in Lemma 2.3 using a bounded number of maps.

### Lemma 2.7

### Proof

Lemma 2.7 follows almost immediately from Lemmas 2.4 and 2.5. We include the proof for completion. Let us start by emphasising \(D_{\beta ,M,n}\subseteq (0,\frac{M}{\beta -1})\) for \(\beta \in (1,\beta _n)\) and so is contained in \([\delta ,\frac{M}{\beta -1}-\delta ]\) for some \(\delta \) depending on *M* and \(\beta \). Now fix \(x\in D_{\beta ,M,n}.\) By Lemma 2.4 there exists a bounded number of transformations that map *x* into \([\frac{k_1}{\beta -1}+\epsilon ,\frac{k_2}{\beta -1}-\epsilon ]\) for some \(\epsilon >0\). Applying Lemma 2.5 to the image of *x* within \([\frac{k_1}{\beta -1}+\epsilon ,\frac{k_2}{\beta -1}-\epsilon ]\) allows us to assert that there exists a bounded number of maps that map this image of *x* into \([\Pi _{\beta }((k_1,k_2^n)^{\infty }),\Pi _{\beta }((k_2,k_1,k_2^{n-1})^{\infty })]\). Hence our \(\eta ^1\) exists. The existence of \(\eta ^2\) follows from an analogous argument. \(\square \)

## 3 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

- 1.
\((\tau _N\circ \cdots \circ \tau _1)(x)\in D_{\beta ,k_1,k_2,n}\) for all \(N\in \mathbb {N},\)

- 2.
\(\Big | |(\tau _i)_{i=1}^N|_{k_1}-p_{k_1}N\Big |\le C \quad \text {for all}\quad N\in \mathbb {N},\)

- 3.
\( \Big | |(\tau _i)_{i=1}^N|_{k_2}-p_{k_2}N\Big |\le C \quad \text {for all}\quad N\in \mathbb {N}.\)

### Proof

The proof of Proposition 3.1 relies on devising an algorithm that yields the desired \(\tau \). 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 \(\tau \) will be constructed by concatenating maps of the form guaranteed by Lemma 2.6, maps from \([\Pi _{\beta }((k_1,k_2^n)^{\infty }),\Pi _{\beta }((k_2,k_1,k_2^{n-1})^{\infty })]\) to itself of the form \(T_{k_2}^l\circ T_{k_1}\), and maps from \([\Pi _{\beta }((k_1,k_2,k_1^{n-1})^{\infty }),\Pi _{\beta }((k_2,k_1^n)^{\infty })]\) to itself of the form \(T_{k_1}^l\circ T_{k_2}\). Also, note that since \(\tau \) will be an element of \(\{T_{k_1},T_{k_2}\}^{\mathbb {N}},\) property (2) is equivalent to property (3). So it suffices to prove property (2).

**Step 1**For \(x\in [\Pi _{\beta }((k_1,k_2,k_1^{n-1})^{\infty }),\Pi _{\beta }((k_2,k_1^n)^{\infty })]\) we have

*M*and \(\beta \). Let \(\tau ^1=(T_{k_2},T_{k_1}^{l_1})\). Note that

**Step 2**At this point we remark that

*M*and \(\beta \). Letting \(\kappa ^2=(\kappa ^1,T_{k_1},T_{k_2}^{l_2})\) we see that (3.3) implies

*j*. Assume \(j^*\) is the smallest such

*j*such that (3.11) occurs. We then let \(\tau ^2=\kappa ^{j^*}\). Importantly, analogues of (3.9) and (3.10) hold for each intermediate \(\kappa \) term. Consequently

*j*such that (3.11) occurs, then we let \(\tau \in \{T_{k_1},T_{k_2}\}^{\mathbb {N}}\) be the infinite sequence we attain as the limit of the \(\kappa ^j\). Since each \(\kappa ^{j}\) is a prefix of \(\kappa ^{j'}\) for any \(j'>j\) the infinite sequence \(\tau \) is well defined. In this case the following holds for each \(j\in \mathbb {N}\)

**Step**\(j+1\) Suppose we have constructed \(\tau ^j\) such that

*L*maps. We then repeatedly map this image of \(\tau ^j(x)\) back into \([\Pi _{\beta }((k_1,k_2,k_1^{n-1})^{\infty }),\Pi _{\beta }((k_2,k_1^n)^{\infty })]\) using maps of the form \(T_{k_1}^{l}\circ T_{k_2}\) where \(n\le l\le n'\). We stop if we observe a change of sign. If we observe a change of sign the sequence we will have constructed is our \(\tau ^{j+1}\). It can be shown that this \(\tau ^{j+1}\) will then satisfy

Proposition 3.1 has the useful consequence that for any \(N\in \mathbb {N},\) the sequence \((\tau _1,\ldots ,\tau _N)\) 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 \(k_1,k_2\in \{0,\ldots ,M\}\) such that \(k_1\ne k_2,\) not necessarily \(k_1<k_2\), let \(\mathbf v _{n,k_1,k_2}=(v_0,\ldots ,v_M)\) where all entries are zero apart from \(v_{k_1}=\frac{n}{n+1}\) and \(v_{k_2}=\frac{1}{n+1}.\) In the following we denote the convex hull of a finite set of vectors by \(\text {Conv}(\cdot ).\)

### Proposition 3.2

### Proof

By the Krein–Milman theorem (see [11]) it suffices to check that the extremal points of \(\Delta _{M,\epsilon }\) are in the convex hull of \(\{\mathbf{v }_{n,k_1,k_2}\}_{k_1,k_2}\). However, for any \(\frac{1}{n+1}\le \epsilon \) we clearly have that \(\mathbf q _{\epsilon ,k_1,k_2}\) is a convex combination of \(\mathbf v _{n,k_1,k_2}\) and \(\mathbf v _{n,k_2,k_1}.\)\(\square \)

Equipped with Propositions 3.1 and 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 \(\beta \in (1,\beta _n)\) and \(\mathbf p \in \Delta _{M,\frac{1}{n+1}}\). By Proposition 3.1 we know that for each \(k_1\ne k_2\) we can construct finite sequences of maps \(\tau _{k_1,k_2}\) of an arbitrary length with frequencies equal to \(\frac{n}{n+1}\) and \(\frac{1}{n+1}\) up to a bounded error term. By Proposition 3.2 we know that \(\mathbf{p }\in \text {Conv}(\{\mathbf{v }_{n,k_1,k_2}\}_{k_1,k_2})\). This proposition guarantees the existence of weights that may be used to construct \(\mathbf p \) from the frequencies of the different \(\tau _{k_1,k_2}\). The problem is that we cannot freely concatenate the \(\tau _{k_1,k_2}.\) We have to travel between the \(D_{\beta ,k_1,k_2,n}\) which introduces an error. However, by Lemma 2.7 this error can always be bounded. Consequently by taking repeatedly larger \(\tau _{k_1,k_2}\) this error becomes progressively more negligible, meaning the limiting sequence we construct will achieve the desired frequency \(\mathbf p \).

### Proof of Theorem 1.2

Fix \(n\in \mathbb {N}\) and \(\mathbf p =(p_0,\ldots ,p_M)\in \Delta _{M,\frac{1}{n+1}}.\) To prove Theorem 1.2 it suffices to show that for any \(\beta \in (1,\beta _n)\) and \(x\in (0,\frac{M}{\beta -1})\) we have \(\mathbf p \in \Delta _{M,\beta }(x)\).

**Step 1**Without loss of generality we may assume that \(x\in D_{\beta ,k_1^1,k_2^1,n}.\) By Proposition 3.1 we can construct \(\tau ^{1,1}\) such that \(\tau ^{1,1}(x) \in D_{\beta ,k_1^1,k_2^1,n}\), \(\tau ^{1,1}\in \{T_{k_1^1},T_{k_2^1}\}^{N_{1,k_1^1,k_2^1}},\)

**Step**\(j+1\) Suppose we have constructed \((\psi ^{i})_{i=1}^j\) and \((\alpha ^{i})_{i=1}^j\) which satisfy

- 1.For each \(1\le i\le j\)and \(\alpha ^{i}(x)\in D_{\beta ,k_1^1,k_2^1,n}.\)$$\begin{aligned} \alpha ^i=(\psi ^1,\ldots ,\psi ^i) \end{aligned}$$
- 2.For each \(1\le i \le j\)$$\begin{aligned} \psi ^i=\left( \tau ^{i,1},\eta ^{i,1},\ldots , \tau ^{i,M(M+1)},\eta ^{i,M(M+1)}\right) . \end{aligned}$$
- 3.
For each \(1\le i\le j\) and \(1\le p\le M(M+1)\) we have \(|\eta ^{i,p}|\le L.\)

- 4.
For each \(1\le i\le j\) and \(1\le p\le M(M+1)\) we have \(\tau ^{i,p}\in \{T_{k_1^p},T_{k_2^p}\}^{N_{i,k_1^p,k_2^p}}.\)

- 5.For each \(1\le i\le j\) and \(1\le p\le M(M+1)\) we haveand$$\begin{aligned} \Big | |(\tau ^{i,p})|_{k_1^p}-\frac{n\cdot N_{i,k_1^p,k_2^p}}{n+1}\Big |\le C \end{aligned}$$(3.20)$$\begin{aligned} \Big ||(\tau ^{i,p})|_{k_2^p}-\frac{N_{i,k_1^p,k_2^p}}{n+1}\Big |\le C. \end{aligned}$$(3.21)

*x*replaced by \(\alpha ^j(x)\) we obtain a sequence

By property (1) above it follows that \(\alpha ^{j}\) is prefix of \(\alpha ^{j'}\) for any \(j'>j\), so the limiting infinite sequence \(\alpha \) is well defined. Clearly \(\alpha \in \Omega _{\beta ,M}(x)\) by property (1). By Lemma 2.1 to prove our theorem it remains to show that \(\alpha \) satisfies the required digit frequency properties.

*k*was arbitrary this completes our proof. \(\square \)

### 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 \((p_k)_{k\in D^c}\) be as in the statement of Theorem 1.4. Under the assumptions of this theorem there exists \(\mathbf q \in \Delta _{M,\frac{1}{n+1}}\) and \(\mathbf q '\in \Delta _{M,\frac{1}{n+1}}\) such that \(q_k\ne q_k'\) for all \(k\in D\), and \(q_k=q_k'=p_k\) for all \(k\in D^c\).

- 1.There exists a sequence \((N_p)\) such thatfor all \(k\in D.\)$$\begin{aligned} \lim _{p\rightarrow \infty }\frac{|(\alpha _l)_{l=1}^{N_p}|_k}{N_p}=q_k \end{aligned}$$
- 2.There exists a sequence \((N_j')\) such thatfor all \(k\in D.\)$$\begin{aligned} \lim _{j\rightarrow \infty }\frac{|(\alpha _l)_{l=1}^{N_j'}|_k}{N_j'}=q_k' \end{aligned}$$
- 3.For all \(k\in D^c\) we have$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{|(\alpha _l)_{l=1}^{n}|_k}{n}=p_k. \end{aligned}$$

To construct the \(\alpha \in \Omega _{\beta ,M}(x)\) 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 \(\mathbf q \). Once we have a sufficiently good approximation to \(\mathbf q \) we change our algorithm to construct an expansion with digit frequencies described by \(\mathbf q ',\) then once we have a sufficiently good approximation to \(\mathbf q '\) we switch back to \(\mathbf q \) and so on. \(\square \)

## 4 Proof of Theorem 1.6

In this section we prove Theorem 1.6. We split the proof into the following two propositions.

### Proposition 4.1

### Proof

### Proposition 4.2

### Proof

## 5 Applications to biased Bernoulli convolutions

*E*is an arbitrary Borel subset of \(\mathbb {R}\). 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 \(\beta \in (1,M+1]\) and \(\mathbf q \in \Delta _M\) such that \(\mu _\mathbf{q }\) 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 et al. [24], and Varju [25].

*x*to be

*x*. We remark that when it comes to calculating \(\underline{d}_{\mu _\mathbf{q }}\) and \(\overline{d}_{\mu _\mathbf{q }}\) it suffices to consider sequences of the form \(r_N:=\frac{M}{\beta ^N(\beta -1)}\).

### Proposition 5.1

### Proof

**q**as above we let

### Corollary 5.2

Similarly, combining Theorem 1.1 and Proposition 5.1 we conclude the following statement.

### Corollary 5.3

## Notes

### Acknowledgements

This research was supported by the EPSRC Grant EP/M001903/1. The author would like to thank Wolfgang Steiner for posing the question that led to this work, and Thomas Jordan for pointing out the applications to biased Bernoulli convolutions. Part of this work was completed whilst the author was visiting the Mittag-Leffler institute as part of the program “Fractal geometry and Dynamics”. The author thanks the organisers and staff for their support. The author would also like to thank the referees for their comments.

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