L q -spectra of self-affine measures: closed forms, counterexamples, and split binomial sums

We study L q -spectra of planar self-affine measures generated by diagonal matrices. We introduce a new technique for constructing and understanding examples based on combinatorial estimates for the exponential growth of certain split binomial sums. Using this approach we disprove a theorem of Falconer and Miao from 2007 and a conjecture of Miao from 2008 concerning a closed form expression for the generalised dimensions of generic self-affine measures. We also answer a question of Fraser from 2016 in the negative by proving that a certain natural closed form expression does not generally give the L q -spectrum. As a further application we provide examples of self-affine measures whose L q -spectra exhibit new types of phase transitions. Finally, we provide new non-trivial closed form bounds for the L q -spectra, which in certain cases yield sharp results.


Introduction and summary of results
The L q -spectrum is an important concept in multifractal analysis and quantifies global fluctuations in a given measure. In the setting of self-affine measures, the L q -spectrum is notoriously difficult to compute, and is only known in some specific cases, see for example [4,5] and in some settings a generic formula is known [1,2,6]. Even in some cases where a formula is known, it is not given by a closed form expression which makes explicit calculations (and theoretical manipulation) difficult. Some attention has been paid to the provision of closed form expressions in [3,5,8] and these works provide the main motivation for this one.
First we consider the setting of Fraser [5] and Feng-Wang [4], where the self-affine measures are generated by diagonal systems. Fraser [5, theorem 2.10] provided closed form expressions for the L q -spectra in many cases, but often required some extra assumptions on the defining system. He asked if these technical assumptions could be removed and if his formula held in general [5, question 2.14]. We answer this question in the negative by providing an explicit family of counterexamples, see theorem 3.8. Despite the fact that the predicted closed form expression does not hold, we are able to provide new, non-trivial, closed form bounds for the L q -spectra, see theorem 3.11. We also provide examples of self-affine measures whose L q -spectra exhibit new types of phase transitions, see theorem 3.9. Specifically, we construct examples where the L q -spectrum is differentiable at q = 1 but not analytic in any neighbourhood of q = 1.
Secondly, we consider the setting of Falconer-Miao [3] and Miao [8] where the self-affine measures are generated by upper triangular matrices. The paper [3] was mainly concerned with dimensions of self-affine sets, but towards the end it states a closed form expression for the generalised q-dimensions (these are a normalised version of the L q -spectra) in a natural generic setting [3, theorem 4.1]. The proof of this result was just sketched and when the result appeared later in Miao's thesis [8, theorem 3.11] the full proof was only given for 0 < q < 1 and the formula only conjectured to hold for q > 1. We show that this formula and conjecture of Miao are false for q > 1 in general by providing an explicit family of counterexamples, see theorem 4.4. We are able to provide new, non-trivial, closed form bounds for the generalised q-dimensions, see theorem 4.5 and also give new conditions which guarantee that the conjectured formula does hold, see corollary 4.6.
A key technical tool is the following growth result for split binomial sums: if one considers the binomial expansion of (1 + x) k , where x > 1 is fixed, and splits the sum in half, then the ratio of the two halves grows exponentially in k, see theorem 2.1.

Preliminaries and split binomial sums
For background on iterated function systems (IFS) see [9]. We recall some basic concepts. Suppose we have an IFS {S i } i∈I consisting of contracting affine transformations of R n where I is some finite index set. Then it follows from Hutchinson's theorem (see for instance [9, theorem 9.1]) that there exists a unique non-empty, compact set F satisfying which we call the self-affine set associated to {S i } i∈I . We shall be interested in measures supported on such sets, and a natural class can be constructed as follows. Suppose we have a self-affine set F given by the IFS {S i } i∈I acting on R n , and a probability vector {p i } i∈I with each p i ∈ (0, 1). Then there exists a unique Borel probability measure μ on R n satisfying which we call the self-affine measure associated to {S i } i∈I and {p i } i∈I . We close this section with a technical result which states that a certain split binomial sum ratio grows exponentially. This result will be used to provide counterexamples later in the paper.
where the limit is taken along odd integers k.
Proof. Fix x > 1 and let k 1 be odd.
It follows that on the one hand and on the other hand

Since 1+x
2 √ x > 1 by the arithmetic-geometric mean inequality the result follows easily.

Diagonal systems and the L q -spectrum
We now turn to the first class of IFS we shall study and introduce the L q -spectrum of the associated self-affine measure. We begin by introducing the necessary background from [5,7].

Definition 3.1 (L q -spectrum).
If μ is a Borel probability measure on R n with support denoted by supp(μ) then for q 0 the upper and lower L q -spectrum of μ are defined to be respectively (where τ μ (1) and τ μ (1) are interpreted as being 0). If these two values coincide we define the L q -spectrum of μ, denoted τ μ (q), to be the common value.
This quantity is of special interest in multifractal analysis due to its relationship with the fine multifractal spectrum. In particular if the multifractal formalism holds then the fine multifractal spectrum of μ is given by the Legendre transform of τ μ (for details see [11]). Definition 3.2 (diagonal system). We say a self-affine IFS is a diagonal system if it is an IFS consisting of affine transformations of R 2 whose linear part is a contracting diagonal matrix.
Note that necessarily the maps that make up diagonal systems are of the form S i (x, y) = T i (x, y) + t i , where T i is a contracting linear map which can be written in matrix form as We shall also assume that our IFS satisfies the following separation condition.
Let I * = k 1 I k denote the set of all finite sequences with entries in I. For i = (i 1 , . . . , i k ) ∈ I * let S i = S i 1 • S i 2 • · · · • S i k and let p(i) = p i 1 p i 2 . . . p i k . Also write α 1 (i) α 2 (i) for the singular values of the linear part of S i and write c(i) = c i 1 c i 2 . . . c i k and d(i) = d i 1 d i 2 . . . d i k . In particular, for all i = (i 1 , . . . , i k ) ∈ I * , α 1 (i) = max{c(i), d(i)} and α 2 (i) = min{c(i), d(i)}. Now define π i : R 2 → R by and subsequently define τ i (q) by τ i (q) := τ π i (μ) (q). Note that by definition τ i (q) is simply the L q -spectrum of the projection of μ| S i (F) onto the longest side of the rectangle S i ([0, 1] 2 ). Furthermore as π i is always equal to π 1 or π 2 , it follows that τ i (q) is always equal to either τ 1 (q) or τ 2 (q). For s ∈ R and q 0, define the q-modified singular value function, ψ s,q : I * → (0, ∞) by and for each k ∈ N define the value Ψ s,q k by It now follows from lemma 2.2 in [5] and standard properties of sub-multiplicative sequences that we may define a function P : It follows from lemma 2.3 in [5] that we may define another function, γ : [0, ∞) → R, by P(γ(q), q) = 1. We shall refer to this function as a moment scaling function. The importance of this function is the following theorem from [5].
Theorem 3.4 [ 5, theorem 2.6] . Suppose that μ is generated by a diagonal system and satisfies the ROSC. Then This tells us that finding a closed form expression for τ μ (q) is equivalent to finding a closed form expression for γ(q).
Note that we may approximate γ(q) numerically by functions γ k (q), where for each k ∈ N we define γ k (q) : In order to find a closed form expression Fraser defined functions The following lemma tells us some useful information about the relationship between γ A , γ B and τ 1 , τ 2 .
Lemma 3.5 [ 5, lemma 2.9] . Let μ be generated by a diagonal system and q 0. Then either This lemma is particularly helpful as it allows us to state Fraser's main result on closed form expressions from [5].
The fact that we only have an inequality involving γ(q) when min{γ A (q), γ B (q)} τ 1 (q) + τ 2 (q), combined with the observation that the above conditions (the sums involving logarithms) do not look especially natural, led Fraser to ask the following question.
By presenting a family of counterexamples we shall answer this question in the negative. In particular we provide a family of diagonal systems consisting of two maps equipped with the Bernoulli-(1/2, 1/2) measure such that

A family of counterexamples
We now give examples answering question 3.7 in the negative. We require a family of measures such that the two conditions in theorem 3.6 fail. At the same time we also need to ensure that they are simple enough to allow us to estimate Ψ s,q k in (3.1) effectively. We prove the following result, which states that, for a certain explicit family of self-affine measures generated by diagonal systems, τ μ (q) is not equal to either γ A (q) or γ B (q) for all q > 1. Theorem 2.1 will be of key importance in establishing this result.
Let μ be the self-affine measure defined by the probability vector (1/2, 1/2) and the diagonal system consisting of the two maps, S 1 and S 2 , where Then, for q > 1, More precisely, for q > 1, γ A (q) = γ B (q) < 0 and, writing s to denote this common value, Proof. Let q > 1. We begin by noting that due to the relative simplicity of the maps we are working with it is straightforward to show that τ 1 (q) = τ 2 (q) = γ A (q) = γ B (q). We shall denote this common value by s, and also note that s < 0. Let k be odd. We may write Ψ s,q k as using the fact that p = 1/2 and s = τ 1 (q) = τ 2 (q). Since the maps S 1 and S 2 commute, we can is the number of times S 1 was used in the composition of S i . For such maps, since c > d, where We now consider the ratio X q k /(1 − X q k ). By our binomial result (theorem 2.1) and the definition of and therefore We may rearrange (and cancel a factor 2 −kq c ks ) to give We note that as c > d and as s < 0 we have (d/c) s > 1. Thus by theorem 2.1, as k → ∞. Thus we also have X q k 1/k → δ as k → ∞. By following similar reasoning we can deduce the same result for Y q k . In particular, (this follows from relabelling the summation by j = k − i and using the fact that k k− j = k j . Note that (3.5) gives exactly the same as the expression we found for X q k /(1 − X q k ) earlier, and so we must also have Y q and by definition of P(t, q) and γ(q) We can upgrade this result to get the stated quantitative upper bound (3.2) by considering the function P(t, q) more closely. For k 1 and i ∈ I k , α 1 (i) (cd) k/2 and therefore, for ε = s − γ(q) > 0, and therefore which proves the theorem.

New examples of phase transitions
Here we record a simple consequence of theorem 3.8 relating to phase transitions. We say that the L q -spectrum τ μ (q) exhibits a first order phase transition at a point t ∈ R if the derivative of τ μ is discontinuous at t. Likewise we say τ μ (q) exhibits an nth order phase transition at t ∈ R if its derivatives up to the (n − 1)th order are continuous at t but the nth order derivative is discontinuous at this point. The differentiability of the L q -spectrum is important and has many interesting consequences. Key among these is the fact that if τ μ (1) exists then its absolute value gives the Hausdorff dimension of the measure in question, see [10]. We can use theorem 3.8 to provide examples of behaviour relating to higher order phase transitions at q = 1. We are unaware of any other method for constructing such examples. Theorem 3.9. There exists a planar self-affine measure μ defined by an IFS satisfying the ROSC such that τ μ , the L q -spectrum of μ, is differentiable at q = 1 but not analytic in any neighbourhood of q = 1.

Proof.
Consider the planar self-affine measures considered in theorem 3.8. As the functions τ 1 , τ 2 are the L q -spectra of the measures π 1 μ, π 2 μ and these measures are self-similar and satisfy the open set condition, it follows that they are real analytic on (0, ∞), see [9, chapter 17], (in particular, they are differentiable at q = 1). We can therefore apply theorem 2.12 in [5] and conclude that the function γ(q) is differentiable at q = 1, so that τ μ = γ is differentiable at q = 1.

New closed form lower bounds
We now know that γ(q) is not in general given by either the maximum or minimum of γ A (q) and γ B (q). However, by developing a quantitative version of the argument in [5] used to prove theorem 3.6 we are able to provide new closed form lower bounds for γ(q) for all planar diagonal systems. Given x ∈ R we write x + to denote the maximum of x and 0. Theorem 3.11. Let μ be a self-affine measure generated by a diagonal system and let q 0. Then In particular, are both strictly less than 1, which ensures that this result provides a strictly better bound than γ(q) τ 1 (q) + τ 2 (q) in the case when γ(q) Proof. We prove that γ(q) L A (q). The inequality γ(q) L B (q) follows by an analogous argument which we omit. Let {θ i } i∈I denote an arbitrary probability vector, and for each k ∈ N, define a number n(k) ∈ N by Note that k − |I| n(k) k. We consider the n(k)th iteration of I and define noting that We also define numbers c, d and p (for which we suppress the dependence on k) by First assume that i∈I c θ i i > i∈I d θ i i . In particular this assumption implies that c > d for k sufficiently large. Indeed and therefore c > d for all Therefore, for all sufficiently large k, i ∈ J k and s ∈ R, By definition of c, d and p we may write this as We now introduce a form of Stirling's approximation which states that for n ∈ N sufficiently large n log n − n log n! n log n − n + log n.
Using this as well as (3.6) we find that for k sufficiently large where the last line follows from the above version of Stirling's formula. Continuing to bound and introducing an exponent of 1/n(k) we get log Ψ s,q n(k) where the last line uses the fact that k − |I| n(k). Taking the limit as k → ∞ the right-hand side tends to If this is non-negative then and therefore γ(q) s. Second, assume that i∈I c θ i i < i∈I d θ i i . In this case, a completely analogous argument proves that if then P(s, q) 1 and so γ(q) s. Finally, if i∈I c θ i i = i∈I d θ i i then we cannot guarantee that c > d or d > c for all k sufficiently large. We can however conclude that we must have either c d or d c (or both) for infinitely many k, so by choosing an appropriate subsequence we can reduce to one of the above two cases. Since we do not know which case we are in (c d or d c) we must require that both of the above summation conditions hold. Putting the above three cases together we have therefore shown that γ(q) sup s : there exists a probability vector {θ i } i∈I such that either In the above we have the freedom to choose a probability vector {θ i } i∈I . A natural choice here, suggested by considering Lagrange multipliers, is to take i i∈I (note that this is indeed a probability vector by definition of γ A ). We now let s = γ A (q) − ε for ε 0. We want to see how small we can make ε (ideally we want ε = 0) such that the two conditions hold simultaneously. The first holds trivially, since For the second to hold, we require Rearranging this, we see that this is equivalent to requiring We note that when Fraser's original condition from theorem 2.10 in [5] holds, namely if then right-hand side of (3.7) is negative so we may take ε = 0. Otherwise we use the bound for ε given in (3.7). Putting these two cases together therefore gives us that Finally we note that so our lower bound is indeed an improvement on γ(q) τ 1 (q) + τ 2 (q) in the case when γ(q) min{γ A (q), γ B (q)}.

An example
Here we present an example of a diagonal system satisfying the assumptions of theorem 3.8 where we take c = 3/4 and d = 1/4, which is displayed in figure 2. We know from theorem 3.8 that τ μ (q) = γ(q) is not given by the maximum or minimum of γ A (q) and γ B (q) for q > 1. It is therefore natural to seek bounds for the L q -spectrum. Let q > 1. Focussing on upper bounds, theorem 3.6 implies that, for q > 1, γ A (q) = γ B (q) = τ 1 (q) = τ 2 (q) = s < 0, where s is the solution of 2 −q c s + 2 −q d s = 1, Figure 1. Graph of our new upper and lower bounds for the L q -spectrum (solid lines), labelled by the theorem they come from. For reference we also show graphs of the previously known upper bound min{γ A (q), γ B (q)} (long dash) and the previously known lower bound τ 1 (q) + τ 2 (q) (short dash), as well as the lower bound 1 − q, which is specific to this setting (dots). Concerning lower bounds, theorem 3.11 implies that We also note a couple of trivial lower bounds. Since γ(0) = 1 (the box dimension of the support of μ), γ(1) = 0, and γ is necessarily convex, it follows that 1 − q is a lower bound for τ μ (q). We also know that τ 1 (q) + τ 2 (q) is a lower bound for τ μ (q), see a remark following [5, question 2.14]. Figure 1 shows a plot of these bounds for q ∈ [1,20]. We see that our new lower bound, max{L A (q), L B (q)} is a strict improvement on the lower bound of 1 − q outside of the range (1.7, 9.3).

Generalised q-dimensions in the generic setting
In [3] Falconer and Miao considered self-affine sets and measures generated by IFSs consisting of upper-triangular matrices. This paper was mainly concerned with dimensions of self-affine sets, but towards the end of the paper they stated a closed form expression for the generalised q-dimensions in the measure setting (here, generalised q-dimensions simply refer to the L qspectrum normalised by 1 − q). We show that in fact their formula does not always hold when q > 1. We begin by recalling some definitions and notation from [3]. For a finite Borel measure μ on R n and q ∈ R, q = 1, Falconer and Miao discuss the generalised q-dimensions of μ, denoted D q (μ). This is simply defined to be the L q -spectra of μ normalised by 1 − q, that is provided the appropriate limits exist. In order to calculate the generalised q-dimensions of self-affine measures μ associated with contracting upper triangular matrices T 1 , . . . , T N and probabilities p 1 , . . . , p N Falconer and Miao studied the quantity d q (T 1 , . . . , T N , μ) defined, for each q 0 (q = 1) to be the unique t satisfying This approach was introduced in [1] where it was shown that for q ∈ (1, 2) the generalised qdimensions of a self-affine measure is generically given by d q (T 1 , . . . , T N , μ) in an appropriate sense. See [2] where further results along these lines were obtained for almost self-affine measures. It is therefore of great interest to provide closed form expressions for d q (T 1 , . . . , T N , μ) or at least to be able to estimate it effectively. We state the result using our notation and only in the planar case, although it is possible to apply our methods to the higher dimensional setting. Let T 1 , . . . , T N denote a collection of contracting non-singular 2 × 2 upper triangular matrices and let c i , d i denote the diagonal entries of the ith matrix. Define a function P 0 : and, for each q ∈ [0, 1) ∪ (1, ∞), let u 0 (q) be defined by P 0 (u 0 (q), q) = 1, provided a solution exists and otherwise simply let u 0 (q) = 2.
Theorem 4.2 [ 3, theorem 4.1] . Let μ be a planar self-affine measure generated by an IFS of upper triangular matrices as above. Then for q ∈ [0, 1) In the paper [3], this result was suggested to hold for all q 0 (q = 1). The result appeared again in Miao's PhD thesis [8, theorem 3.11] in which he noted that, in fact, he could only establish the result for q ∈ [0, 1). Miao conjectured that the result should still hold for q > 1, see discussion leading up to [8, theorem 3.11]. Our main result in this section, which is essentially an analogue of theorem 3.8 adapted to this situation, proves that theorem 4.2, does not hold for q > 1 in general.

A family of counterexamples relating to generalised q-dimensions
Before considering the range q > 1 we note that a better lower bound than u 0 (q) is available simply by changing the maximum to a minimum in the definition of P 0 , which is natural for q > 1. We define P * 0 : [0, 2] × [0, 1) ∪ (1, ∞) → [0, ∞) by P * 0 (t, q) = P 0 (t, q) for q ∈ [0, 1) and for q > 1 by Let u(q) be defined by P * 0 (u(q), q) = 1, provided a solution exists and otherwise simply let u(q) = 2. Note that u(q) = u 0 (q) for q ∈ [0, 1) and u(q) u 0 (q) for q > 1 with strict inequality a possibility. This inequality comes from the fact that the functions that we are taking the maximum or minimum of are increasing in t for q > 1. We expect that when conjecturing a closed form expression for d q (T 1 , . . . , T N , μ) for q > 1, Miao [8] was thinking of u(q) rather than u 0 (q).

Lemma 4.3.
For all q 0 (q = 1) we have Proof. It suffices to only consider the range q > 1 since for q < 1 this result is covered by [3,8]. Write α 1 (i) α 2 (i) for the singular values of the matrix T i . Firstly suppose that 0 u(q) < 1 and therefore where we have used the fact that c i and d i are multiplicative in i. Therefore, for t = u(q), and since the expression on the left is increasing in t (since q > 1) If 1 u(q) < 2, then the proof follows similarly noting We leave the details to the reader.
Despite this simple improvement on the lower bound, we prove that d q (T 1 , . . . , T N , μ) is still not generally equal to u(q) for q > 1. Let μ be the self-affine measure defined by the probability vector (1/2, 1/2) and the diagonal system consisting of the two maps, T 1 and T 2 , defined by For q > 1 let u(q) be defined by P * 0 (u(q), q) = 1, that is, u(q) is the unique solution of Then, for all q > 1, More precisely, for all q > 1, Proof. We adapt the proof of theorem 3.8. Let q > 1, k be odd, and consider the following sum noting that u(q) u(0) 1. As before we see that for i ∈ I k if T 1 appears i times in the composition of T i and T 2 appears k − i times, then, since c > d, and so the above is equal to We again define X q k and Y q k to be the left and right terms of this expression. Continuing with exactly the same approach as in the proof of theorem 3.8 and applying theorem 2.1, where in this case x = (c/d) u(q)(q−1) > 1, we find that Recall that since 1 − q < 0, it follows in this setting that is a strictly increasing function of t and therefore as required. We can upgrade this result to get the stated quantitative lower bound by considering the definition of d q (T 1 , T 2 , μ) more closely. For k 1 and i ∈ I k , we have α 1 (i) (cd) k/2 and therefore, for ε = d q (T 1 , and therefore which proves the theorem.

New closed form bounds for generalised dimensions
Despite the fact that d q (T 1 , . . . , T N , μ) is not given by the value predicted by Falconer-Miao [3,8] q > 1, we can still find upper bounds in the case when our matrices are diagonal by following the approach of section 3.3. To simplify notation and aid readability, we only pursue such bounds in the planar case but higher dimensional analogues could be proved similarly. For convenience here we let I denote the set {1, . . . , N}. We also let t 1 , t 2 , s 1 , s 2 be defined by the following equations: and, as in the previous section, define u(q) by P * 0 (u(q), q) = 1. We may assume that u(q) < 2, as otherwise there is nothing to prove, and we note that u(q) is always equal to one of t 1 , t 2 , s 1 , s 2 . Once again we write x + for the maximum of x ∈ R and 0. Theorem 4.5. Let μ be a self-affine measure generated by a diagonal system in R 2 and assume that q > 1.  . Previously this formula was only known for 0 < q < 1, see [3,8]. Middle: plot of the first condition from corollary 4.6, which is satisfied for the whole range of q. Right: plot of the second condition from corollary 4.6, which is not satisfied. Observe that the value at q = 0 gives the affinity dimension of the support of our measure, which in this case is 1. Also recall that, by Falconer's result [1, theorem 6.2], the generalised qdimensions of μ are given by d q (T 1 , T 2 , T 3 , μ) for 1 < q 2 almost surely upon if randomising the translation vectors, provided the norms of the matrices are strictly less than 1/2. This is displayed in figure 4.