Measures on the Spectra of Algebraic Integers
aa r X i v : . [ m a t h . D S ] F e b Measures on the Spectra of Algebraic Integers
Alex Batsis and Tom Kempton ∗ February 16, 2021
Abstract
Given a real number β >
1, the spectrum of β is a well studied dy-namical object. In this article we show the existence of a certainmeasure on the spectrum of β related to the distribution of randompolynomials in β , and discuss the local structure of this measure. Wealso make links with the question of the Hausdorff dimension of thecorresponding Bernoulli Convolution. Given a real number β > A , the spectrum X A ( β ) := ( n X i =1 c i β n − i : n ∈ N , c i ∈ A ) has been the focus of much attention. In particular, when A = { , · · · , ⌊ β ⌋} then it is known that X A ( β ) is uniformly discrete if and only if β is a Pisotnumber (i.e. an algebraic number, all of whose Galois conjugates have mod-ulus strictly less than one) [3, 6, 9, 12]. Additionally, X A ( β ) is relativelydense in this setting, making the sets X A ( β ) Delone sets (uniformly discrete,relatively dense). Delone sets give useful mathematical models for quasicrys-tals and so the above construction gives a number-theoretic construction ofimportant physical objects. ∗ The University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom. X A ( β )[7, 10, 14]. If β is a Pisot number then X A ( β ) can be generated by a substi-tution system [10]. Moreover, for Pisot β there is a naturally related cut andproject set which contains X A ( β ). In all known examples of Pisot β with A ⊂ Z the set X A ( β ) coincides with this cut and project set, but the ques-tion of whether these sets always coincide remains open, and there are someexamples with a complex alphabet for which the cut and project set containsfinitely many extra points which are not in X A ( β ) [14]. A generalisationof this cut and project structure to general hyperbolic algebraic integers isgiven in section 4.We are interested in measures on the sets X {− , , } ( β ). In particular, we areinterested in what one can say about the measures µ n given by µ n ( x ) = 14 n N n ( x )where N n ( x ) = { a · · · a n , b · · · b n ∈ { , } n : n X i =1 ( a i − b i ) β n − i = x } . The measure µ n is the distribution of the set of differences n X i =1 a i β n − i − n X i =1 b i β n − i where each a i , b i is picked from { , } according to the ( , ) Bernoulli mea-sure. We focus on the case that β is an algebraic integer and a root ofa { -1,0,1 } polynomial but does not have any Galois conjugates of absolutevalue one, we call such β hyperbolic. There has been a lot of recent research into a different class of measures (Pattersonmeasures) on cut and project sets. These are related to diffraction on quasicrystals, wherethey play the role of the intensity of the Bragg peak [21, 22]. Loosely speaking, thedifference between the class of measures that we study and Patterson measures is that ourmeasures incorporate information on the number of different codings a · · · a n for which P ni =1 a i β n − i = x , whereas Patterson measures do not. The analogue of µ n ( x ) for thePatterson measure would be (more or less) γ n ( x ) = { ( y, z ) ∈ ( X { , } ( β )) : y − z = x } . This difference is crucial for our applications. µ n , appro-priately rescaled, have a limit µ as n tends to infinity, and whether that limithas any ‘local structure’ analagous to that of the set X A ( β ). Assuming sometechnical (but checkable) conditions, our results hold for general hyperbolic β , but all of the ideas behind our proofs are present in the golden mean case,which is notationally much simpler, and for this reason we prove our resultsfirst for the golden mean. The golden mean also has the advantage that thehigher dimensional objects which we construct are only two dimensional, andso can be more easily visualised.Our main theorems are the following. Theorem 1.1.
Let β be hyperbolic. Then there exists a real number λ > ,such that for all x ∈ X ( β ) the limit measure µ given by µ ( x ) := lim n →∞ λ n N n ( x ) exists and has µ ( x ) ∈ (0 , ∞ ) for x ∈ X ( β ) . Furthermore, the measure µ hasinfinite total mass. In the case that β has other Galois conjugates of absolute value larger thanone, we prove this theorem by lifting to a measure ¯ µ supported on a higherdimensional Delone set, whose projection onto the first coordinate gives µ .Our second theorem gives an explicit way to calculate µ ( x ) using any codeof x . Theorem 1.2.
Let β be hyperbolic. There exist a natural number k , a × k vector W , and three k × k matrices M − , M and M such that for any x ∈ X ( β ) and c · · · c n ∈ {− , , } n with x = P ni =1 c i β n − i , µ ( x ) = 1 λ n ( W M c · · · M c n ) . Here ( W M c · · · M c n ) denotes the first entry of the row vector W M c · · · M c n . In fact the vector
W M c · · · M c n also holds information on the values of µ ( y )for other values of y ∈ X ( β ). There is a set of translations d , · · · , d k ∈ R ,with d = 0, such that, for x = P ni =1 c i β n − i , µ ( x + d i ) µ ( x ) = ( W M c · · · M c n ) i ( W M c · · · M c n ) . µ to calculate its values at different points. We cando this, but we need first to replace the dependence of µ ( x ) on the codingof x with a dependence on the position of a point x c corresponding to x inthe ‘contracting space’. To describe this, we must first describe a geometricconstruction related to β -expansions in algebraic bases.Let β have Galois conjugates β · · · β d of absolute value larger than one andGalois conjugates β d +1 · · · β d + s of absolute value smaller than one. Definethe contracting space K c by K c = F d +1 × F d +2 × · · · × F d + s where F k = R if β k ∈ R , F k = C if β k ∈ C \ R . Then, for i ∈ {− , , } define the contraction S i on K c by S i ( x d +1 , · · · , x d + s ) = ( β d +1 x d +1 + i, · · · , β d + s x d + s + i ) . The maps { S − , S , S } form an iterated function system on K c with anattractor that we denote R . This is a standard construction in numera-tion/tiling theory, although it is more usual to consider a sub-IFS using onlythose codes which correspond to greedy β -expansions [1]. To each point x = P ni =1 c i β n − i there exists a corresponding point in the contracting space: x c = n X i =1 c i ( β n − id +1 , β n − id +2 , · · · , β n − id + s ) = S c n ◦ · · · S c (0) ∈ R . It is important to stress that the point x c corresponding to x is independent ofthe coding c , · · · , c n of x , this holds since β d +1 · · · β d + s are Galois conjugatesof β . Theorem 1.3.
Assume that Condition 4.1 holds. There exists a set ∆ =( v , · · · v k ) of translations such that for any j ∈ { · · · k } there is a function f j : R → R such that for any x ∈ X ( β ) with x + v j also in X ( β ) we have ln (cid:18) µ ( x + v j ) µ ( x ) (cid:19) = f j ( x c ) . Furthermore any x ∈ X ( β ) can be reached from by applying a finite num-ber of translations from ∆ . There exists a word w and constants C > , C ∈ (0 , such that for any a · · · a n ∈ {− , , } n which contains r non-overlapping copies of the word w , f j varies by at most C C r − on S a ◦ · · · ◦ S a n ( R ) . f j gives rise to the following continuityproperties of f j .1. Continuity almost everywhere:
For any fully supported ergodicmeasure ν on R , each f j is continuous ν -almost everywhere2. Continuity at most lattice points:
For any fully supported measure m on {− , , } and any ǫ > n ∈ N and D ⊆ {− , , } n such that m n ( D ) > − ε and | f j ( x ) − f j ( y ) | < ε for all x, y ∈ X ( β ) with x c , y c ∈ S a ◦ · · · ◦ S a n ( R ) for any a · · · a n ∈ D .These latter two continuity properties follow since ν almost every sequencecontains infinitely many copies of the word w , and that for any r and any ǫ > n such that a proportion at least 1 − ǫ of {− , , } wordsof length n contain r non-overlapping occurences of w .We use this theorem extensively in our follow up article. For now, we limitour application of this theorem to the golden mean case, where we show thatthe values of µ ( x ) can be obtained via a cocycle over an interval exchangetransformation on R = ( − φ , φ ), see Theorem 3.3.In Section 2 we describe some links with the dimension theory of Bernoulliconvolutions, which allows us to state some new conjectures about Bernoulliconvolutions. In Section 3 we prove Theorems 1.1, 1.2 and 1.3 in the specialcase that β is the golden mean. Finally in Section 4 we prove these theoremsfor the general case of hyperbolic β . Our interest in the measures µ measures stems from a link with the study ofthe dimension and possible absolute continuity of Bernoulli convolutions ν β ,defined below. We describe here connections with dimension theory for Pisotnumbers, links between our work and the question of absolute continuity of5 β for non-Pisot hyperbolic β are postponed to a follow up article, in whichwe generalise [19] to give a condition for the absolute continuity of ν β interms of the growth of µ n ([ − β − , β − ]), which in turn can be stated in termsof rapid equidistribution to Lebesgue measure of the measures µ n | [ − β − , β − ] .We then use the local structure of the measures µ n described in Theorem 1.3and an analogue of Theorem 3.3 to study this equidistribution.Given a number β ∈ (1 , ν β is the weak ∗ limit ofthe measures ν β,n given by ν β,n = X a ··· a n ∈{ , } n n δ P ni =1 a i β − i where δ x denotes the Dirac probability measure on x . The measure ν β isa probability measure on [0 , β − ] and is perhaps the simplest example of aself-similar measure with overlaps. The question of whether ν β is absolutelycontinuous for some given parameter β goes back to Jessen and Wintner[18]. Erd˝os showed that ν β is singular when β is a Pisot number [8], andindeed Garsia showed that such Bernoulli convolutions have dimension lessthan one [12]. There has been very substantial progress on the dimensiontheory of Bernoulli convolutions in the last decade, stemming from the workof Hochman [17], and in particular it is now known that non-algebraic β giverise to Bernoulli convolutions of dimension one [25], whereas for algebraic β there are algorithms to determine whether or not ν β has dimension one [5, 2].For a summary of recent research into the dimension theory of BernoulliConvolutions see [24].There have been many numerical studies into the dimensions of BernoulliConvolutions associated with Pisot numbers. The evidence we have suggeststhat for Pisot numbers of large degree the dimension of the correspondingBernoulli convolution is close to one [2, 13, 15, 16, 20]. We formalise thisconjecture here. Conjecture 1.
Let β n be a sequence of Pisot numbers in the interval (1 , and suppose that the degree of β n tends to infinity as n → ∞ . Then dim H ( ν β n ) → . We have not seen this conjecture formally stated before, but it seems consis-tent with the (admittedly fairly limited) numerical evidence that we have.6he rest of this section is devoted to giving another conjecture on the mea-sures µ n and showing that this new conjecture would be sufficient to proveConjecture 1.It was proved in Hochman [17] that, for algebraic β the dimension of theBernoulli convolution ν β is given bydim H ( ν β ) = min (cid:26) , H ( β )log( β ) (cid:27) . Here the Garsia entropy H ( β ) is given by H ( β ) := lim n →∞ n H n ( β )where H n ( β ) = − X a ··· a n ∈{ , } n n log n { b · · · b n ∈ { , } n : n X i =1 ( a i − b i ) β n − i = 0 } ! . As noted in [2], one can use Jensen’s inequality to reverse the order of thesummation and the log, to get H n ( β ) ≥ − log n { a · · · a n , b · · · b n ∈ { , } n : n X i =1 ( a i − b i ) β n − i = 0 } ! = log(4 n ) − log( N n (0)) . In particular, our main theorem, Theorem 1.1, introduces a constant λ equalto the exponential growth rate of N n (0), using this constant we get H ( β ) ≥ log(4) − log λ. (1)Our contribution here in the Pisot case is to link the question of how closeto being equidistributed µ is to the value of λ , broadly when µ | [ − β − , β − ] iswell distributed with respect to Lebesgue measure then Equation 1 gives alower bound for the dimension of ν β which is close to one. Our approach hereis more or less that of trying to understand something about the maximaleigenvalue of a matrix by studying the corresponding eigenvector. We usethe following elementary lemma from linear algebra.7 emma 2.1. Let M be a k × k matrix with maximal eigenvalue ρ and asso-ciated left eigenvector V = ( v , · · · , v k ) normalised so that P ki =1 v i = 1 . Let r i := P kj =1 M i,j denote the i th row sum of M . Then ρ = k X i =1 v i r i . Let β be a Pisot number and I β := [ − β − , β − ]. Then, as noted before, λ counts the (weighted) growth of the number of words in {− , , } n forwhich P ni =1 c i β n − i = 0, the weighting comes from giving each word weight2 m where m is the number of occurences of letter 0 in the word. Whenever P ni =1 c i β n − i = 0 we have that P mi =1 c i β m − i is in the interval I β , and so is in X ( β ) ∩ I β which is a finite set V = { v , · · · , v k } thanks to the Garsia Sep-aration Property [11]. We write down a matrix M indexed by { v , · · · , v k } with ( M ) i,j = v j = βv i ± v j = βv i . Then the measure µ I β := µ ( I β ) µ | I β gives mass to v j equal to the jth entry ofthe left probability eigenvector of M associated with maximal eigenvalue λ .Furthermore, we can read off the i th row sum r i of M (associated to point v i ∈ X ( β ) ∩ I β ) immediately, since we need only know which of βv i − , βv i and βv i + 1 lie in I β .Let the function g β : I β → { , , , } be given by g β ( x ) = χ I β ( βx −
1) + 2 χ I β ( βx ) + χ I β ( βx + 1) . Then r j = g β ( v j ) and so by Lemma 2.1 we have λ = X v j ∈ V g β ( v j ) µ I β ( v j ) = Z I β g β ( x ) dµ I β ( x ) . (2)A short calculation gives that if L I β denotes normalised Lebesgue measureon I β then Z I β g β ( x ) d L I β ( x ) = 4 β . We have the following theorem. 8 heorem 2.1.
Let β n be a sequence of Pisot numbers and suppose that W ( µ I βn , L I βn ) → where W denotes the Wasserstein metric on the space of probability measureson the Euclidean line. Then dim H ( ν β n ) → .Proof. The function g β is a step function on I β and it is straightforward togive an upper bound for | µ I β ( A ) − L I β ( A ) | for any of the intervals A uponwhich the step function is constant in terms of the distance between µ I β and L I β . These upper bounds are uniform in β . This in turn yields uniform upperbounds on R I β g β dµ I β , and so by equation 2 we have a uniform upper boundon λ ( β ) − log( β ) in terms of W ( µ I βn , L I βn ).Finally, for Pisot β n dim H ( ν β n ) = H ( β n )log( β n ) ≥ log 4 − log λ ( β n )log β n → log 4 − log (cid:16) β n (cid:17) log( β n ) = 1 . as required.The matrix M ( β ) associated to a Pisot number β is very large for β oflarge degree, and so the numerical evidence we have is limited, but the evi-dence that we have does suggest that the measures µ I βn are increasingly wellequidistributed for sequences β n of Pisot numbers in (1 , − ǫ ) with degreetending to infinity, see Table 1. The ǫ here is to exclude the multinacci family,which has different behaviour .Finally, we give our conjecture on the distribution properties of the measures µ I βn . A proof of this conjecture would imply that Conjecture 1 is true byTheorem 2.1. Conjecture 2.
Let ǫ > and let ( β n ) be a sequence of Pisot numbers in theinterval (1 , − ǫ ) such that the degree of β n tends to infinity as n tends toinfinity. Then the distance d ( µ I βn , L I βn ) → as n → ∞ , and consequently, by Theorem 2.1, dim H ( ν β n ) → . Many structures related to the multinacci family β nn − β n − n − · · · − β n , are well understood. β Bound W ( µ β , Leb) Matrix Size x − x − x − x − x − x − x − x − x − x − x − x − x − x − x − x − x − x − x − x − x − x − x − x − x − x + 1 1.7785 0.995758 0.0246573 951 x − x − x − x − x − x − x − x − x − x + x − x − x − x − x − x − x − x − β ∈ (1 ,
2) of degree less than six, together with theWasserstein distance to normalised Lebesgue measure. Multinacci numbers,which have somewhat different behaviour, are in bold.
In this section we prove our main theorems for the special case that β isequal to the golden mean φ . Throughout we use the maps T i : R → R givenby T i ( x ) = φx + i .Recall that X ( φ ) = X {− , , } ( φ ) = ( n X i =1 c i φ n − i : n ∈ N , c i ∈ {− , , } ) and that, for x ∈ X ( φ ), N n ( x ) := { a · · · a n , b · · · b n ∈ { , } n : n X i =1 ( a i − b i ) φ n − i = x } We give the special case of Theorem 1 . β = φ . Theorem 3.1.
There exists a number λ > such that limit lim n →∞ λ n N n ( x ) =: µ ( x )10 xists for each x ∈ X ( φ ) . Here λ is easily computed as the maximal eigenvalue of a finite matrix M defined below. This theorem will be proved as part of the proof of Theorem3.2.There are several ways to describe the measure µ . One could construct aninfinite transition matrix corresponding to dynamics on X ( φ ) induced bythe maps T , T , T − such that the values of µ ( x ) correspond to entries ofthe eigenvector corresponding to the maximal eigenvalue. In particular, forany finite K we can describe µ | X ( φ ) ∩ [ − K,K ] by reading off the values of aneigenvector of a finite matrix. We give instead a harder construction whichallows us to see local structure in the measure µ . Lemma 3.1.
There exist matrices M , M , M − , each of dimensions × such that for any x = P ni =1 c i φ n − i ∈ X ( φ ) we have N n ( x ) = ( M c · · · M c n ) , Proof.
This proof is similar to the proof of Lemma 3.1 in [2], we are justusing a larger digit set.If x = P ni =1 c i φ n − i for some word c · · · c n ∈ {− , , } n then we start bytracking words d · · · d n ∈ {− , , } n such that n X i =1 c i φ n − i = X i =1 d i φ n − i , i.e. n X i =1 ( c i − d i ) φ n − i = 0 . (3)Here d i represents a difference a i − b i where a i , b i ∈ { , } , and so whencounting words we want to double count the case d i = 0 since it correspondsboth to a i = b i = 1 and a i = b i = 0. This accounts for the 2 in the definitionof the matrices M , M , M − .Now the equality 3 is equivalent to T c n − d n ◦ · · · ◦ T c − d (0) = 0 , (4)11here each c i − d i ∈ {− , − , , , } . The maps T i are expanding, and inparticular if x ≥ φ then T i ( x ) ≥ φ , and if x ≤ − φ then T i ( x ) ≤ − φ , forany i ∈ {− , − , , , } . Thus if equation 3 holds then for each m ≤ n wehave T c m − d m ◦ · · · ◦ T c − d (0) ∈ ( − φ, φ ) . By the Garsia separation lemma, or by direct calculation, one can show thatthere are a finite number of points of the form T c m − d m ◦ · · · ◦ T c − d (0) whichlie in ( − φ, φ ) when c i , d i ∈ {− , , } . In fact there are 17 such points, wecall the set of such possible values V = { v , · · · , v } with v = 0.Now in general the difference c i − d i can take values in {− , − , , , } , butif we know the value of c i then c i − d i can only take three of these values, if c i = 1 then c i − d i can take values 0 1 or 2 for example.Let M be the 17 ×
17 matrix with rows and columns indexed by elementsof V , with ( M ) ij = v j = T ( v i ) or v j = T − ( v i )2 v j = T − ( v i )0 otherwiseThis is the transition matrix for the maps T c i − d i where we know c i = 1 and d i ∈ {− , , } , the values 1 and 2 occur because we have one way of letting d i = a i − b i equal 1 or − d i = 0.Similarly, let M − be the matrix with rows and columns indexed by elementsof V , with ( M − ) ij = v i = T ( v j ) or v i = T ( v j )2 v i = T ( v j )0 otherwiseand let M be the matrix with rows and columns indexed by elements of V ,with ( M ) ij = v i = T ( v j ) or v i = T − ( v j )2 v i = T ( v j )0 otherwise . Then given c , · · · c n ∈ {− , , } n , the ( i, j )th term of the matrix M c n · · · M c represents the number of d · · · d n ∈ {− , , } for which T c n − d n ◦ · · · T c − d ( v i ) = v j . (5)12gain here when we refer to the ‘number’ of d · · · d n we are double countingwhen d i = 0 because we have two ways of putting a i − b i = 0.Thus in order to count equalities of the form (4), we need to use (5) with v i = v j = v = 0 . We conclude that the number of a · · · a n , b · · · b n such that P ni =1 ( a i − b i ) φ n − i = x is given by the top left entry of the matrix M c n · · · M c ,where c · · · c n is any {− , , } code for which x = P ni =1 c i φ n − i .We now state and prove Theorem 1.2 for the special case that β is equal to φ . Theorem 3.2.
Let W be the left eigenvector of M corresponding to themaximal eigenvalue λ . Then for any x = P ni =1 c i φ n − i ∈ X ( φ ) we have µ ( x ) = 1 λ n ( W M c M c · · · M c n ) , that is, λ n µ ( x ) is the first entry in the × vector W M c · · · M c n .Proof. In the previous lemma we showed how to count the number of words a , · · · a n , b · · · b n with P ni =1 ( a i − b i ) φ i = x , given knowledge of one code c · · · c n ∈ {− , , } n such that x = n X i =1 c i φ n − i . (6)Here it was important that the length of the word c · · · c n coding x corre-sponded with the N n which we want to calculate. But if equation 6 holdsthen it is also true that x = n X i =1 c i φ n − i + 0 φ n + 0 φ n +1 + · · · + 0 φ n +( k − . So again using Lemma 3.1 we see that N n + k ( x ) = ( M k M c · · · M c n ) , = (1 0 0 · · · ) M k M c · · · M c n . λ is the maximal eigenvalue of M then, since M is primitive, there existsa corresponding eigenvector W such that1 λ k (1 0 0 · · · ) M k → W Putting the previous equations together gives that if x = P ni =1 c i φ n − i then µ ( x ) = lim k →∞ λ n + k N n + k ( x )= lim k →∞ λ k λ n (1 0 0 · · · ) M k M c · · · M c n = 1 λ n W M c · · · M c n . It is also important to note that if x = P ni =1 c i φ n − i then the vector λ n W M c · · · M c n doesn’t just hold information on µ ( x ), which is the first entry, but also holdsinformation on the values of µ at other elements of X ( φ ). Lemma 3.2.
For v k the kth element of V we have µ ( x + v k ) = 1 λ n ( W M c M c · · · M c n ) k , that is, λ n µ ( x + v k ) is the k th entry in the × vector W M c · · · M c n .Proof. This follows directly from the proof of the previous lemma and equa-tion 5.This allows us to start to discuss local structure for µ . We want to describehow one can use dynamics to move through the measure µ and write downthe set of pairs { ( x, µ ( x )) : x ∈ X ( φ ) } . To do this, we must first recall thecut and project structure of the set X ( φ ).14 .1 The Structure of X ( φ ) The work of this subsection is well known to experts. We first show that set X ( φ ) can be dynamically generated. One can move from a level- n sum to alevel-( n + 1) sum in the construction of X ( φ ) by observing that n +1 X i =1 c i φ n +1 − i = φ n X i =1 c i φ n − i ! + c n +1 . Thus with T i ( x ) := φx + i as before we see that X ( φ ) = { T c n ◦ · · · ◦ T c (0) : n ∈ N , c i ∈ {− , , }} . (7)As φ = φ + 1 we can consider multiplication by φ in terms of its action onnumbers of the form z φ + z . We let π e : Z → R be given by π e (cid:18) z z (cid:19) := z φ + z and π c : Z → R be given by π c (cid:18) z z (cid:19) := − φ z + z . We will later refer to π e as projection in the expanding direction and π c as projection in the contracting direction. Note that π e : Z → R and π c : Z → R are injective (if they were not then x − x − φ ).Then φ (cid:18) π e (cid:18) z z (cid:19)(cid:19) = z φ + z φ = ( z + z ) φ + z = π e (cid:18)(cid:18) (cid:19) (cid:18) z z (cid:19)(cid:19) and so T i : X ( φ ) → X ( φ ) lifts to a map ˜ T i : Z → Z given by˜ T i (cid:18) z z (cid:19) = (cid:18) (cid:19) (cid:18) z z (cid:19) + (cid:18) i (cid:19) . We let ˜ X ( φ ) := (cid:26) ˜ T c n ◦ · · · ◦ ˜ T c (cid:18) (cid:19) : n ∈ N , c i ∈ {− , , } (cid:27) X ( φ ) around the origin, with expanding and contractingeigenvectors shownand have the relation X ( φ ) = π e ( ˜ X ( φ )).One can study the structure of X ( φ ) directly on the real line, this was donefor example in [10] where the substitution structure of X ( φ ) was described.However, some properties of X ( φ ) are easier to see if we first study thestructure of ˜ X ( φ ). For example, from equation (7) we see that the uniformlydiscrete set X ( φ ) is a subset of the dense set { z φ + z : z , z ∈ Z } , but itis not immediately apparent which values of ( z , z ) correspond to points in X ( φ ).Lifting to ˜ X ( φ ) the structure becomes clear. The matrix (cid:18) (cid:19) has oneexpanding eigenvector and one contracting eigenvector, and the maps ˜ T i canbe described in terms of their action on points written in terms of theseeigenvectors.Note that if π c (cid:18) z z (cid:19) = x then π c ( ˜ T i (cid:18) z z (cid:19) ) = − xφ + i =: S i ( x ) . Then the system { S , S , S − } is a contracting iterated function system withattractor [ − φ , φ ], and so for any point (cid:18) z z (cid:19) = ˜ T a n ◦· · · ˜ T a (cid:18) (cid:19) ∈ X ( φ )we have π c (cid:18) z z (cid:19) = S a n ◦ · · · S a (0) ∈ ( − φ , φ ). The converse is also trueand is contained in the following lemma.16 emma 3.3. The set ˜ X ( φ ) consists of all pairs (cid:18) z z (cid:19) ∈ Z for which π c (cid:18) z z (cid:19) lies in the interval ( − φ , φ ) .Furthermore, if π c (cid:18) z z (cid:19) ∈ S d ◦ · · · S d k ( − φ , φ ) for some d , · · · , d k ∈{− , , } k then for all sufficiently large n there exists a word c · · · c n + k ∈{− , , } n + k with c n + k · · · c = d · · · d k and such that (cid:18) z z (cid:19) = ˜ T c n + k ◦ · · · ◦ T c (cid:18) (cid:19) Proof.
One inclusion was proved in the paragraph before the statement ofthis lemma.Now let ( z , z ) ∈ Z have π c ( z , z ) ∈ ( − φ , φ ). We wish to find a word c · · · c n such that (cid:18) z z (cid:19) = ˜ T c n ◦ · · · ˜ T c (cid:18) (cid:19) , or equivalently (cid:18) (cid:19) = ˜ T − c ◦ · · · ˜ T − c n (cid:18) z z (cid:19) . (8)We first observe that for any (cid:18) z z (cid:19) with π c (cid:18) z z (cid:19) ∈ ( − φ , φ ) and π e (cid:18) z z (cid:19) ∈ [ − φ, φ ] one can find words c · · · c n such that Equation 8 holds. Since thereare only finitely many pairs (cid:18) z z (cid:19) in this bounded region one can checkthis observation with a finite calculation.Now let (cid:18) z z (cid:19) have π c (cid:18) z z (cid:19) ∈ ( − φ , φ ), but place no restriction on π e (cid:18) z z (cid:19) ∈ ( − φ, φ ). By the IFS construction of the contracting interval,we can choose arbitrarily long words i · · · i n ∈ {− , , } such that ˜ T − i n ◦· · · ˜ T − i ( (cid:18) z z (cid:19) ) still has contracting coordinate in the interval ( − φ , φ ).But since inverse maps ˜ T − i contract the expanding direction, the expanding17oordinate will eventually lie in [ − φ, φ ], and by the previous paragraph weknow that we can return to (cid:18) (cid:19) . Finally we not that if we had π c (cid:18) z z (cid:19) ∈ S d ◦ · · · S d k ( − φ , φ ) then we can choose the word i · · · i n to start with d · · · d k .It is worth stressing that the first three quarters of the preceeding proofgeneralises easily to any algebraic integer β , but the finite check that anyinteger pair suitably close to the origin can return to the origin under themaps ˜ T − i needs verifying for each β and we don’t know that it is alwaystrue.One interesting consequence of Lemma 3.3 is that in order to understand thedistance from some point ˜ T c n ◦ · · · ˜ T c (cid:18) (cid:19) to its close neighbours in ˜ X ( φ ),we need only to know about π c ( ˜ T c n ◦ · · · ˜ T c (cid:18) (cid:19) ).Given x ∈ X ( φ ) let ˜ x denote the corresponding point in ˜ X ( φ ) and let x c = π c (˜ x ). For K ∈ R let x ∈ X ( φ ). Call the set( X ( φ ) − x ) ∩ [ − K, K ] = { y − x : y ∈ X ( φ ) , y − x ∈ [ − K, K ] } the K -neighbourhood of x . Lemma 3.4. [Local Structure for X ( φ ) ] For any K > there exists a fi-nite partition of ( − φ , φ ) such that the K -neighbourhood of any x ∈ X ( φ ) depends only upon which partition element of ( − φ , φ ) x c lies in.Proof. This follows from the analagous statement for ˜ X ( φ ), which has a fairlydirect proof following Lemma 3.3, since one needs only to consider whichtranslations in Z can be performed without leaving the contracting windowor moving by a distance of more than K in the expanding direction.Finally, we outline how to use dynamics to describe the odometer map whichmaps x ∈ X ( φ ) to min { y ∈ X ( φ ) : y > x } . For hyperbolic non-Pisot β we will also require that expansions of Galois conjugatesare close to the origin, see section 4. d : X ( φ ) → R + denote the distance from x ∈ X ( φ ) to min { y ∈ X ( φ ) : y > x } . That is, let d be defined by d ( x ) = min { y ∈ X ( φ ) : y > x } − x. Proposition 3.1.
The odometer map x → x + d ( x ) on X ( φ ) lifts to theskew-product map O : X ( φ ) × X c ( φ ) → X ( φ ) × X c ( φ ) by ˜ d ( x, x c ) = ( x + 2 φ − , x c − φ − x c ∈ [ φ, φ ]( x + φ − , x c − − φ ) x c ∈ (0 , φ )( x + 2 − φ, x c + 2 + φ ) x c ∈ [ − φ , O on the contracting direction is of auniquely ergodic interval exchange transformation. Proof.
The fact that there is some partition of ( − φ , φ ) telling us how toevolve a skew-product map which is a lift of d follows immediately fromLemma 3.4 with K = φ −
1. It is a finite calculation to write down the mapexactly. µ Proposition 3.1 dealt with how one can move locally through the set X ( φ )using only knowledge on the position in the contracting direction, we wantto build a similar theorem which also incorporates knowlede of the values µ ( x ), we do this by building a cocycle over the odometer map O .Given x ∈ X ( φ ) let x c denote the corresponding point in the contractingwindow ( − φ , φ ). We recall from Lemma 3.3 that for x ∈ X ( φ ) and for anyword d · · · d k , x can be written x = P ni =1 c i φ n − i with c n − k +1 · · · c n = d k · · · d if and only if x c ∈ S d ◦ · · · ◦ S d n ( − φ , φ ).Now let us map real 1 ×
17 vectors U with strictly positive first entry ontothe corresponding projective space by letting( U ′ ) i = ( U ) i +1 ( U ) for (1 ≤ i ≤ x = P ni =1 c i φ n − i ∈ X ( φ ) the corresponding vector V ( x ) = ( W M c M c · · · M c n ) ′ considered as an19lement of real projective space. To be concrete, we define the 1 ×
16 vector V ( x ) by ( V ( x )) i = ( W M c M c · · · M c n ) i +1 ( W M c M c · · · M c n ) = µ ( x + v i ) µ ( x ) . It follows from the proofs of the previous two statements that these vectorsdo not depend on the choice of code c · · · c n of x . We can also write V ( x )as a function V ( x c ) of the position in the contracting window.Consider the metric d on the space of 1 ×
16 non-negative vectors by letting d ( U, V ) = max i ∈{ , ··· } | ln( v i ) − ln( u i ) | . Two vectors
U, V are at infinite distance from one another if there exist i, j ∈ { · · · } such that u i = 0 v i = 0 or v i = 0 u i = 0. Lemma 3.5.
Suppose that A is a × matrix with A , > such that forany pair of parameters ( i, j ) ∈ { , · · · , } one of the following holds1. ( i, j ) is in a zero row, i.e. ( A ) i ′ ,j = 0 for all i ′ ∈ { , · · · , } ( i, j ) is in a zero column, i.e. ( A ) i,j ′ = 0 for all j ′ ∈ { , · · · , } ( A ) i,j > .Then there exists a constant C < such that, for any × vectors U, V with positive first entries and with d ( U ′ , V ′ ) < ∞ we have d (( U A ) ′ , ( V A ) ′ ) < Cd ( U ′ , V ′ ) . Furthermore, there exists
K > such that, for any any × vectors U, V with positive first entry (and possibly with d ( U ′ , V ′ ) = ∞ ), d (( U A ) ′ , ( V A ) ′ ) < K. This lemma is proved carefully in section 4.
Lemma 3.6.
The matrix M satisfies the condition of Lemma 3.5. This can be verified by a short calculation.20ne can also see that given a 17 ×
17 non-negative matrix B withstrictlypositive top left entry and two 1 ×
17 vectors U and V with strictly positivefirst entries, d (( U A ) ′ , ( V A ) ′ ) ≤ d ( U ′ , V ′ ) . This shows that matrices M , M and M − do not expand distances betweenvectors in our metric.Finally we are able to state Theorem 1.3 in the special case that β = φ anddealing only with nearest neighbours. Recall that, for x ∈ X ( φ ), d ( x ) :=min { y − x : y ∈ X ( φ ) , y > x } . Proposition 3.2.
For x ∈ X ( φ ) with corresponding point x c ∈ ( − φ , φ ) define f ( x c ) by ln( µ ( x + d ( x ))) − ln( µ ( x )) = f ( x c ) . Then f is bounded and is continuous at each x c ∈ X c ( φ ) except for and φ . If we defined d ′ on ( φ , φ ) by d ′ ( x c ) := d ( x ) then 0 and φ are the points in( φ , φ ) where d ′ ( x c ) is not continuous. Proof.
We have already shown that d ( x ) = φ − x c ∈ [ φ, φ ) φ − x c ∈ (0 , φ )2 − φ x c ∈ ( − φ , φ − , φ − − φ correspond to entries v k of V . Then by Lemma 3.2 we see that f ( x c ) := ln( µ ( x + d ( x ))) − ln( µ ( x ))appears as the log of a ratio of two entries in the vector ( W M c · · · M c n ) forany c · · · c n coding x . Since both x and x + d ( x ) have strictly positive mass,the difference of the logs is finite so f ( x c ) ∈ R .We now discuss the continuity properties of f . Let x ∈ X ( φ ) and ǫ > K and C be the quantities introduced in Lemma 3.5 associatedto M , and let r ∈ N be such that KC r − < ǫ . Let c · · · c n be a code of x containing at least r copies of the word 0000000, this can be done for exampleby taking any expansion of x and adding lots of zeros to the start.21ow x c is contained in the interval S c n ◦ S c n − ◦· · ·◦ S c ( − φ , φ ). Let y ∈ X ( φ )be another point with y c ∈ S c n ◦ S c n − ◦ · · · ◦ S c ( − φ , φ ). Then y can bewritten y = P md =1 d i φ m − i for some code d · · · d m with d m − n · · · d n = c · · · c n ,as in Lemma 3.3.Assume that x c and y c lie in the same one of the intervals ( − φ , , (0 , φ ),[ φ, φ ) so that d ( x ) = d ′ ( x c ) = v j . Then | f ( x c ) − f ( y c ) | = | ln( W M c · · · M c n ) j − ln( W M d · · · M d m ) j | = | ln( W M c · · · M c n ) j − ln( W M d · · · M d m − n − M c · · · M c n ) j |≤ d (( W M c · · · M c n ) ′ , ( W M d · · · M d m − n − | {z } =: U M c · · · M c n ) ′ )= d (( W M c · · · M c n ) ′ , ( U M c · · · M c n ) ′ ) ≤ KC r − < ǫ. Here the final line follows since c · · · c n contains r non-overlapping occurencesof the word M , the first of which guarantees that d (( W M c · · · M c n ) ′ , ( U M c · · · M c n ) ′ ) < K and the subsequent r − C , thanksto Lemmas 3.5 and 3.6.We have now completed the proofs of analogues of Theorems 1.1, 1.2, and1.3 in the special case of the golden mean, although the analogue of 1.3 wedid only for moving to nearest neighbours.Putting everything together, we get the following theorem which demon-strates how one can move through the measure µ on X ( φ ), and how onecould start to study it using ergodic theory. Theorem 3.3.
Let the map ψ : X ( φ ) × ( − φ , φ ) × R be given by φ ( x, y, z ) = ( x + 2 φ − , y − φ − , z + f ( y )) y ∈ [ φ, φ )( x + φ − , y − φ − , z + f ( y )) y ∈ (0 , φ )( x + 2 − φ, y + 2 + φ , z + f ( y )) y ∈ ( − φ , Then if x is the n th element to the right of in X ( φ ) we have that ( x, x c , µ ( x )) = ψ n (0 , , . Thus we have that many of the properties of µ can be studied by studying ψ , which is really a skew-product over an interval exchange transformationon the contracting window ( φ , φ ). 22 Measures on the spectra of general hyper-bolic algebraic integers
In this section we show how to extend the previous work to general hyperbolicalgebraic integers and prove Theorems 1.1, 1.2 and 1.3. As stated in theintroduction, the motivation is to study measures of the form µ n ( x ) = 14 n { a · · · a n , b · · · b n ∈ { , } n : n X i =1 ( a i − b i ) β n − i = x } . Given β , we lift µ n to a measure ¯ µ n living on a lattice subset of a multidimen-sional euclidean space K . We prove that there is λ > n ¯ µ n /λ n converges to a measure ¯ µ . We also prove that there are local patterns inthe measure ¯ µ that repeat in a way that we understand. This means thatwe understand how the measure of a lattice point changes when we moveto nearby points on the lattice . In particular there is a non-trivial linearsubspace K c of K such that the following holds. Under conditions and givena suitable vector d then for typical x the ratio ¯ µ ( x + d )¯ µ ( x ) is determined, up tocertain accuracy, by the approximate position of the orthogonal projectionof x on K c . That is the numbers of the form ¯ µ ( x + d )¯ µ ( x ) are approximately equalfor all x projecting on to the same small region of K c .Let β = β ∈ (1 ,
2) be an algebraic integer with Galois conjugates β , ..., β d , β d +1 , ..., β d + s such that | β | , ..., | β d | > | β d +1 | , ..., | β d + s | ∈ (0 , β n = ( β n , ..., β nd + s ). For this section we let T i ( x , ..., x d + s ) = ( β x + i, ..., β d + s x d + s + i ) , these maps are higher dimensional lifts of their analogues in the previoussection. For Galois conjugates β i ∈ C let F β i = R if β i ∈ R and F β i = C if β i ∈ C \ R . We define the sets We don’t state an analogue of Theorem 3.3 for the higher dimensional case since thereis no natural choice of ‘next point’ to move to when we are working in higher dimensionalEuclidean space. One could state such results, perhaps by identifying a strip which isinfinite in only one direction and describing the dynamics to move through such a strip. := d + s Y i =1 F β i , K c := { } d × F β d +1 × ... × F β d + s , ¯ Z := { a d + s − ¯ β d + s − + ... + a ¯ β : a d + s − , ..., a ∈ Z } , and ¯ X ( β ) := ( n X i =1 a i ¯ β n − i : n ∈ N , a ..., a n ∈ {− , , } ) = { T a n ◦ ... ◦ T a (0) : n ∈ N , a ..., a n ∈ {− , , }} where 0 denotes the origin in K .The set ¯ Z is a lattice in K ∼ = R P d + si =1 dim( F βi ) . That is because { ¯ β , ..., ¯ β d + s − } is an independent subset of the real vector space K . That can be checkedusing the formula for the determinant of the Vandermonde matrix. It isuseful to keep in mind that for each i ∈ Z we have T i ( ¯ Z ) ⊆ ¯ Z , in particular¯ X ( β ) ⊆ ¯ Z .Notice that all coordinate projections, restricted on ¯ Z , are injective so there isin a sense a natural identification of ¯ Z to any image of it under a coordinateprojection. Here by a coordinate projection we mean any map from K toitself, of the form ( a , ..., a d + s ) ( a κ , ..., a d + s κ d + s ) where κ , ..., κ d + s ∈{ , } . As in the one dimensional case, we define the measure ¯ µ n on ¯ Z by¯ µ n ( x ) = 14 n ¯ N n ( x )where¯ N n ( x ) = ( ( a , ..., a n , b , ..., b n ) ∈ { , } n : n X i =1 a i ¯ β n − i − n X i =1 b i ¯ β n − i = x ) , for x ∈ ¯ Z . It is immediate that ¯ µ n ( ¯ Z \ ¯ X ( β )) = 0, that ¯ µ n ( x ) = µ n ( x ) and¯ N n ( x ) = N n ( x ). We set π c ( x , · · · , x d + s ) = ( x d +1 , ..., x d + s )24o be the projection onto the contracting directions, and S i := ( π c ◦ T i ) | K c .The maps S i are contractions. Let R be the attractor of the overlappingiterated function scheme { S − , S , S } . We have immediately that π c ( ¯ X ( β )) = π c { T a n ◦ ... ◦ T a (0) : n ∈ N , a ..., a n ∈ {− , , }} = { S a n ◦ ... ◦ S a (0) : n ∈ N , a ..., a n ∈ {− , , }} ⊂ R since 0 ∈ R . Definition 4.1.
Let a = ( a , ..., a n ) ∈ {− , , } n . We define [ a ] := S a ◦ ... ◦ S a n ( R ) . Finally we define a set of small differences between points in X ( β ). Definition 4.2.
Let ∆ = { x − y : x, y ∈ X ( β ) and ∃ c · · · c n , d · · · d n ∈ {− , , } n : T c n ◦ · · · T c ( x ) = T d n · · · T d ( y ) } . That is, ∆ is the set of differences between points x, y ∈ X ( β ) which can bemapped to the same point in the future by the application of maps T i . ∆ isfinite, we write ∆ = { v , · · · , v k } with v = 0.In this section we prove Theorems 1.1, 1.2 and 1.3 by proving higher dimen-sional analogues. In particular, in subsection 4.1 we prove that, for some λ >
0, the measure ¯ µ n λ n converges to an infinite stationary measure ¯ µ (Propo-sition 4.1, which has Theorem 1.1 as a direct corollary.In subsection 4.2 we define matrices A − , A , A playing the role of M − , M , M of the Golden mean example. Given a point x = T a n ◦ ... ◦ T a (0), where a i ∈ {− , , } , we use the matrix A a · ... · A a n to compute the measure¯ µ locally around x (Proposition 4.1), which has Theorem 1.2 as a directcorollary.Finally in subsection 4.3 we show that information about the position of π c ( x ) determines the last few elements a κ , ..., a n of a code of x . This allowus to use arguments involving a modified Birkhoff metric on the product A a · ... · A a n to estimate the local measure around x based on informationabout π c ( x ). This gives rise to Proposition 4.5, which has Theorem 1.3 as acorollary, as explained directly after the proof of Proposition 4.5.25 .1 The limit measure ¯ µ We will denote the vector space of signed measures on ¯ Z by M ( ¯ Z ). For ν ∈ M ( ¯ Z ) we set || ν || = X x ∈ ¯ Z | v ( x ) | . There is a recursive way to go from ¯ µ n to ¯ µ n +1 which gives a dynamicaldescription of ¯ µ n .¯ µ n +1 ( x ) = ( ( a , ..., a n +1 , b , ..., b n +1 ) ∈ { , } n : n +1 X i =1 a i ¯ β n +1 − i − n +1 X i =1 b i ¯ β n +1 − i = x ) = ( ( a , ..., a n +1 , b , ..., b n +1 ) ∈ { , } n +1) : T a n +1 − b n +1 n X i =1 a i β n − i − n X i =1 b i β n − i ! = x ) = X ( a,b ) ∈{ , } ( ( a , ..., a n , b , ..., b n ) ∈ { , } n : n − X i =1 a i β n − i − n − X i =1 b i β n − i = T − a − b ( x ) ) = X ( a,b ) ∈{ , } ¯ µ n ( T − a − b ( x )) . Definition 4.3.
We define the operator L on M ( Z ) by letting ( L ( ν ))( A ) := X ( a,b ) ∈{ , } ν ( T − a − b ( A )) . for A ⊂ Z . Then ¯ µ n satisfies ¯ µ n = L n ¯ µ . Lemma 4.1.
For all n ∈ N and y ∈ ¯ X ( β ) we have ¯ µ n ( y ) ≤ ¯ µ n (0) .Proof. This follows from the Cauchy-Schwarz inequality. Define µ ′ n ( x ) = ( a , ..., a n ∈ { , } n : n X i =1 a i ¯ β n − i = x )
26y the construction of ¯ µ n we have that¯ µ n ( y ) = X x ∈ ¯ Z µ ′ n ( x ) µ ′ n ( x + y ) ≤ X x ∈ ¯ Z µ ′ n ( x ) ! / X x ∈ ¯ Z µ ′ n ( x + y ) ! / ≤ X x ∈ ¯ Z µ ′ n ( x ) ! / X x ∈ ¯ Z µ ′ n ( x ) ! / = X x ∈ ¯ Z µ ′ n ( x ) = X x ∈ ¯ Z µ ′ n ( x ) µ ′ n ( x )= ¯ µ n (0)Now we prove that the measure ¯ µ exists. To do this, we show that it existson arbitrarily large neighbourhoods of the origin. Let I β i ( R ) = (cid:16) − R || β i |− | , R || β i |− | (cid:17) , β i ∈ R \{− , } n z ∈ C : | z | < R || β i |− | o , β i ∈ { z ∈ C : | z | 6 = 1 } \ R ,B β ( R ) = Π d + si =1 I β i ( R ), and ¯ X R ( β ) := ¯ X ( β ) ∩ B β ( R ) . Observe that T i ( ¯ X ( β ) \ ¯ X R ( β )) ⊆ ¯ X ( β ) \ ¯ X R ( β )for R ≥ i ∈ {− , , } . This means that, for R > x ∈ ¯ X R ( β ), anyword a · · · a n for which T a n ◦· · · T a (0) = x has that all the intermediate orbitpoints T a m ◦ · · · T a (0) for m < n also lie in ¯ X R ( β ). Thus, for x ∈ ¯ X R ( β ) wecan compute ¯ N n ( x ) just by studying the dynamics of the maps T i restrictedto ¯ X R ( β ). 27ince ¯ X R ( β ) is a bounded subset of a lattice, it is finite, we enumerate itselements { x , · · · x k R } with x = 0. Then we write down the matrixΛ R ( i, j ) = T ( x i ) = x j or T − ( x i ) = x j T ( x i ) = x j . which encodes the dynamics on ¯ X R ( β ) given by the maps T i . Then since¯ N n ( x j ) counts the number of length n orbit pieces from 0 to x j under themaps T , T , T − , double counting for each use of T , we see that¯ N n ( x j ) = (Λ nR ) ,j . From T i ( ¯ X ( β ) \ ¯ X ( β )) ⊂ ¯ X ( β ) \ ¯ X ( β ) we get that the irreducible componentof Λ R that contains the zero point is contained in ¯ X ( β ) so by lemma 4.1 wehave that the spectral radius of Λ R is equal to the spectral radius of Λ forall R > Definition 4.4.
We set λ := ρ (Λ ) . Now if we knew that the matrices Λ R were irreducible, the existence of µ would be immediate. As it is we require the following lemma, the proof ofwhich is postponed to the appendix. Lemma 4.2.
Let A be a non-negative N × N matrix and e = (1 , , , ..., ∈ R N . Assume thati) A (1 , > ,ii) there exists n ∈ N such that e A n is stricly positive,iii) e A n ( i ) ≤ e A n (1) for all n ∈ N and i ∈ { , ..., N } ,then lim n →∞ e A n /ρ ( A ) n exists. Now by the construction of Λ R and by Lemma 4.1 and Lemma 4.2 we havethe following proposition. 28 roposition 4.1. For each x ∈ ¯ X ( β )¯ µ ( x ) := lim n →∞ ¯ N n ( x ) λ n exists, defining a measure ¯ µ ∈ M ( ¯ Z ) . We conclude this section with three lemmas showing that the measure µ isinvariant under L , that λ <
4, and that the total mass of the measure µ isinfinite. Lemma 4.3. L ¯ µ = λ ¯ µ Proof.
For all x ∈ ¯ X ( β ) we have L ¯ µ ( x ) = ¯ µ ( T − − ( x )) + 2¯ µ ( T − ( x )) + ¯ µ ( T − ( x ))= lim n →∞ λ n (cid:0) ¯ µ n ( T − − ( x )) + 2¯ µ n ( T − ( x )) + ¯ µ n ( T − ( x )) (cid:1) = lim n →∞ λ n L ¯ µ n ( x )= λ lim n →∞ λ n +1 ¯ µ n +1 ( x )= λ ¯ µ ( x ) . For sets X , measures ν ∈ M ( X ) and measurable sets A ⊂ X we let ν | A besuch that ν | A ( B ) = ν ( A ∩ B ) for all measurable B ⊂ X . Lemma 4.4. λ < Proof.
It is clear that if ν ∈ M ( ¯ Z ) is such that || ν || < ∞ then || Lν || = 4 || ν || . L (¯ µ ) = λ ¯ µ and L (cid:0) ¯ µ | ¯ X ( β ) (cid:1) | ¯ X ( β ) = λ ¯ µ | X ( β ) , but (cid:12)(cid:12)(cid:12)(cid:12)(cid:0) L (cid:0) ¯ µ | ¯ X ( β ) (cid:1)(cid:1) | ¯ Z \ ¯ X ( β ) (cid:12)(cid:12)(cid:12)(cid:12) > X ( β ) is not invariant under the maps T , T , T − . Then4 || ¯ µ | ¯ X ( β ) || = (cid:12)(cid:12)(cid:12)(cid:12) L (cid:0) ¯ µ | ¯ X ( β ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:0) L (cid:0) ¯ µ | ¯ X ( β ) (cid:1)(cid:1) | ¯ X ( β ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:0) L (cid:0) ¯ µ | ¯ X ( β ) (cid:1)(cid:1) | ¯ Z \ ¯ X ( β ) (cid:12)(cid:12)(cid:12)(cid:12) = λ || ¯ µ | ¯ X ( β ) || + (cid:12)(cid:12)(cid:12)(cid:12)(cid:0) L (cid:0) ¯ µ | ¯ X ( β ) (cid:1)(cid:1) | ¯ Z \ ¯ X ( β ) (cid:12)(cid:12)(cid:12)(cid:12) > λ || ¯ µ | ¯ X ( β ) || giving us λ < Proposition 4.2. || ¯ µ || = ∞ , i.e., the measure ¯ µ is infinite.Proof. For n ∈ N we get || ¯ µ || = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ n L n ¯ µ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ n L n (cid:0) ¯ µ | { } (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 4 n λ n ¯ µ (0) . The result follows since λ <
4, ¯ µ (0) > n was arbitrary. Let ∆ = { v , · · · , v k } with v = 0. We introduce a k × k matrix withrows/columns corresponding to the points in ∆. Definition 4.5.
For i ∈ {− , , } let A i be the k × k matrix such that ( A i ) m,n = if ∃ j ∈ {− , } : T j − i ( v m ) = v n if T − i ( v m ) = v n otherwise . The matrices A i describe the evolution of local measure as we move from x to T i ( x ), as described in Lemma 4.5. Recall that v = 0 , v , · · · v k are theelements of ∆ (Definition 4.2. We define a vector which describes the localmeasure around x . 30 efinition 4.6. We let v ( x ) = ( µ ( x ) , µ ( x + v ) , · · · , µ ( x + v k )) . Lemma 4.5.
Let x ∈ ¯ X ( β ) . Then λ v ( x ) A i = v ( T i ( x )) . Proof.
We show that( ¯ N n ( x ) , ¯ N n ( x + v ) , · · · , ¯ N n ( x + v k )) A i = ( ¯ N n +1 ( T i ( x )) , ¯ N n +1 ( T i ( x )+ v ) , · · · , ¯ N n +1 ( T i ( x )+ v k )) , the result will follow from this statement.Note that¯ N n +1 ( T i ( x )+ v l ) = ¯ N n ( T − ( T i ( x )+ v l ))+ ¯ N n ( T − − ( T i ( x )+ v l ))+2 ¯ N n ( T − ( T i ( x )+ v l ))(9)where of course ¯ N n ( y ) = 0 for y ¯ X ( β ).Secondly we note that T j ( x + v m ) = T j ( x ) + T ( v m )= T i ( x ) + T ( v m ) + j − i = T i ( x ) + T j − i ( v m ) , which is equal to T i ( x ) + v l if and only if T j − i ( v m ) = v l .So we can rewrite equation 9 to get¯ N n +1 ( T i ( x ) + v l ) = X m ∈{ , ··· ,k } ¯ N n ( x + v m ) χ T − i ( v m )= v l + X m ∈{ , ··· ,k } ¯ N n ( x + v m ) χ T − − i ( v m )= v l + 2 X m ∈{ , ··· ,k } ¯ N n ( x + v m ) χ T − i ( v m )= v l . which is precisely the l th entry of ( ¯ N n ( x ) , ¯ N n ( x + v ) , · · · , ¯ N n ( x + v k )) A i . Proposition 4.1.
Set W = v (0) = ( µ (0) , µ ( v ) , · · · , µ ( v k )) . Let x = P ni =1 c i β n − i .Then v ( x ) = 1 λ n ( W A c · · · A c n ) . n particular, ¯ µ ( x ) = 1 λ n ( W A c · · · A c n ) , i.e. the first entry of the × k vector λ n W A c · · · A c n .Proof. This follows immediately from the previous lemma by writing x = T a n ◦ T a n − ◦ · · · ◦ T a (0) . Since the one dimensional measure µ is the projection of ¯ µ onto the firstcoordinate, Theorem 1.2 follows as a direct corollary to Proposition 4.1. Recall that R is the attractor of the IFS { S − , S , S } and that π c ( ¯ X ( β )) ⊆R . We will assume the following condition. Condition 4.1. ¯ X ( β ) ∩ cl ( B β (1)) = ¯ Z ∩ π − c ( R o ) ∩ cl ( B β (1))This is similar to a condition appearing in Corollary 4.5 of [14]. Here R o denotes the interior of the set. Condition 4.1 is a condition about two finitesets being equal, and so can be easily checked. In words, the condition saysthat a finite patch around zero of the set ¯ X ( β ), which is a higher dimensionalanalogue of the spectrum of β , can be written as a patch of a cut and projectset with window R o . Condition 4.1 implies that the whole set ¯ X ( β ) canbe written as a cut and project set, this is the content of Corollary 4.1. Inevery example we have checked with β ∈ (1 ,
2) a hyperbolic algebraic unitand alphabet A = {− , , } , Condition 4.1 does indeed hold, but thereare examples of Hare, Mas´akov´a and V´avra [14] using complex alphabets inwhich the cut and project set contains extra points. Lemma 4.6.
For each i ∈ {− , , } we have T − i ( ¯ Z ) ⊆ ¯ Z . roof. We need only show that for x = P d − i =0 z i β i where z , · · · , z d − ∈ Z we have that there exist z ′ , · · · z ′ d − such that xβ = P d − i =0 z ′ i β i . Once we haveshown this for x, the corresponding results for the Galois conjugates followdirectly.The result holds because, for β to be a root of a {− , , } -polynomial, it isnecessary that the final term a of the minimal polynomial of β is ±
1. Thenwe use 0 = a d β d + a d β d − + · · · + a β + a = ⇒ β = a d − a β d − + · · · + a − a . and since each of the terms a i − a are integers, since a = ±
1, we have thatdividing by β keeps numbers within the integer lattice as required. Proposition 4.3.
Suppose that x ∈ ¯ X ( β ) has π c ( x ) ∈ [ ε , ..., ε n ] o for some ε , ..., ε n ∈ {− , , } n . Then, under condition 4.1, there are a , ..., a κ ∈{− , , } such that T ε ◦ ... ◦ T ε n ◦ T a κ ◦ ... ◦ T a (0) = x. Recall that [ ε , · · · , ε n ] is a subset of R defined in Definition 4.1, and that[ ε , · · · , ε n ] o is its interior. Proof.
By the iterated function system construction of R , the fact that π c ( x ) ∈ [ ε , · · · , ε n ] gives the existence of arbitrarily long words a , · · · a m ∈{− , , } m such that π c ( x ) ∈ S ε ◦ ... ◦ S ε n ◦ S a ◦ ... ◦ S a m ( R ) . This implies that there is y ∈ ¯ Z with π c ( y ) ∈ R such that x = T ε ◦ ... ◦ T ε n ◦ T a ◦ ... ◦ T a m ( y ) , the fact that y ∈ ¯ Z follows using Lemma 4.6 using that x ∈ ¯ Z . Now x =( x · · · , x d , x d +1 , · · · x d + s ) where the maps T i are expanding on the first d The fact that β is a root of a {− , , } -polynomial isn’t enough to imply that the minimal polynomial of β has digits only {− , , } , but it does follow that the largest andsmallest terms in the minimal polynomial are ± s coordinates. Hence the maps T − i contract the first d coordinates and for any ǫ >
0, for large enough m , thepoint y = ( T ε ◦ ... ◦ T ε n ◦ T a ◦ ... ◦ T a m ) − ( x )must have its first d coordinates within distance ǫ of the box Π di =1 I β i (1). Butsince these points lie in a uniformly discrete set, the first d coordinates mustactually lie in the closure of this box.The final s coordinates must be in R o , since π c ( x ) ∈ S ε ◦ ... ◦ S ε n ◦ S a ◦ ... ◦ S a m ( R o ). Thus( T ε ◦ ... ◦ T ε n ◦ T a ◦ ... ◦ T a m ) − ( x ) ∈ Z ∩ π − c ( R ) ∩ B β (1) , and so by Condition 4.1 there exists b · · · b k ∈ {− , , } k such that( T ε ◦ ... ◦ T ε n ◦ T a ◦ ... ◦ T a m ) − ( x ) = T b ◦ · · · ◦ T b k (0) ∈ ¯ X ( β ) . Then x = T ε ◦ ... ◦ T ε n ◦ T a ◦ ... ◦ T a m ◦ T b ◦ · · · T b k (0)as required. Corollary 4.1.
Under condition 4.1, ¯ X ( β ) = ¯ Z ∩ π − c ( R o ) . This is just the statement of the previous proposition with ε , · · · ε n beingthe empty word. A similar statement appears as Corollary 4.5 in [14]. Lemma 4.7.
Let i, j ∈ { , · · · , k } . Then there exists c , ..., c n ∈ {− , , } such that ( A c · ... · A c n ) ij > . Proof.
The definition of ∆ means there exist a · · · a m ∈ {− , − , , , } m and a m +1 · · · a n ∈ {− , − , , , } such that T a m ◦ · · · ◦ T a ( v i ) = 0 and T a m +1 ◦ · · · ◦ T a n (0) = v j . Then choosing c · · · c m such that a i − c i ∈ {− , , } for each i the result follows directly from the definition of A i .The following lemma is important in defining for us a ‘mixing word’ a n · · · a ∈{− , , } n . 34 roposition 4.4. There is a word w = w , ..., w n ∈ {− , , } n and I, J ⊆ ∆ such that I, J and ( A w · ... · A w n ) i,j = 0 ⇔ i ∈ I or j ∈ J .Proof. We start by building a set I and a word w , · · · , w m such that the i th row of A w . · · · .A w m is a zero row for i ∈ I and ( A w . · · · .A w m ) i, > i ∈ {− , , } , ( A i ) , > v is in ∆, and from the definition of ∆ and lemma 4.7there exist w · · · w m ∈ {− , , } such that( A w · · · A w m ) , > A w · · · A w m is a zero row, in whichcase we declare v ∈ I , or there exists v p ∈ ∆ with ( A w · · · A w m ) ,p >
0. Asin step 2, since v p ∈ ∆ choose a word w m +1 · · · w m such that( A w m · · · A w m ) p, > . Then the product of matrices A w · · · A w m has that entry (3 ,
1) is positive.Furthermore, entry (2 ,
1) is still positive, since A w · · · A w m had entry (2 , w · · · w m k and a set I ⊂ ∆ suchthat the i th row of A w . · · · .A w mk is a zero row for i ∈ I and ( A w . · · · .A w mk ) i, > A T , A T , A T − also have top left entry strictly posi-tive and that for any i ∈ { , · · · k } there exists a word c · · · c n such that( A c · · · A c n ) ( i, >
0. So we repeat the above procedure for the matrices A T , A T , A T − to create a word w ′ · · · w ′ n k and a set J such that the j th row of A Tw ′ · · · A Tw ′ nk is a zero row for j ∈ J , and ( A Tw ′ · · · A Tw ′ nk ) ( j, > A w ′ nk · · · A w ′ hasa set J of zero columns, and for all other columns the first entry is strictlypositive.Now setting w · · · w n = w · · · w m k w ′ n k · · · w ′ we see that the product A w · · · A w n has a set I of zero rows, a set J of zero columns, with all other entries strictlypositive as required. 35 efinition 4.7. Let the mixing word w = w , ..., w n and A w = A w · ... · A w n where w , ..., w n are as in Proposition 4.4 Recall that we defined the 1 × k vectors v ( x ) = ( µ ( x ) , µ ( x + v ) , · · · , µ ( x + v k ))where ∆ = ( v , · · · , v k ) with v = 0. Map the space of 1 × k vectors withpositive first entry onto projective space by letting ( V ′ ) i = ( V ) i +1 ( V ) for 1 ≤ i ≤
16, giving v ′ ( x ) = (cid:18) µ ( x + v ) µ ( x ) , µ ( x + v ) µ ( x ) , · · · µ ( x + v k ) µ ( x ) (cid:19) As before, define the projective distance by d ( U, V ) = max i ∈{ , ··· ,k − } | ln(( V ) i ) − ln(( U ) i ) | ∈ [0 , ∞ ] . Here ln(0) − ln(0) should be understood to take value 0. Proposition 4.2.
There exist C > and C ∈ (0 , such that for any × k vectors U, V , • d ( U A w , V A w ) < C • if d ( U, V ) < ∞ then d ( U A w , V A w ) < C d ( U, V ) . • if d ( U, V ) < ∞ then d ( U A i , vA i ) < d ( U, V ) for any i ∈ {− , , } . If A w was a strictly positive matrix, this would be a standard result ofBirkhoff [4]. It is a simple modification to extend this to the matrices A w ,which are strictly positive on some block with all entries outside of this blockzero. Details of this proof are given in the first author’s thesis. Proposition 4.5.
Assume that Condition 4.1 holds. Then there exist pos-itive constants C , C such that for any word a · · · a r ∈ {− , , } n and forany x, y ∈ ¯ X ( β ) with π c ( x ) , π c ( y ) ∈ [ a ] o , d ( v ′ ( x ) , v ′ ( y )) < C C d ( a ) − where d ( a ) is the number of disjoint occurences of w in a = a · · · a n . roof. By Lemma 4.3 we have that x and y both have expansions endingwith the word a , i.e. we can write x = P ni =1 c i β n − i , y = P mi =1 d i β m − i whereboth c · · · c n and d · · · d m end in word a r · · · a .Then by Lemma 4.5 we can write v ( x ) = 1 λ n v (0) A c · · · A c n = 1 λ n v A c · · · A c n − r | {z } := U A a r · · · A a and v ( y ) = 1 λ n v (0) A d · · · A d m = 1 λ n v A d · · · A d m − r | {z } := V A a r · · · A a But now a r · · · a contains d occurences of the mixing word w . the first ofwhich contracts the distance between vectors U and V to at most C , andthe final d ( a ) − C , as inProposition4.2. Then we have the required result.We note that Theorem 1.3 follows as a direct corollary to Propsition 4.5, asthe vector v ′ ( x ) can be written v ′ ( x ) = (exp( f ( x c )) , exp( f ( x c )) , · · · exp( f k ( x c )))and that d ( v ′ ( x ) , v ′ ( y )) < C C d ( a ) − implies that for each i ∈ { , · · · , k } thedifferences | ln( f i ( x c )) − ln( f i ( y c ) | < C C d ( a ) − . Projecting ¯ µ and the elementsof ∆ onto their first coordinates we are done.Finally we show that all elements of ¯ X can be reached from 0 by applyingfinitely many translations from the set ∆. Lemma 4.8.
Let a , ...a m ∈ {− , , } be such that a ¯ β m − + ... + a m − ¯ β + a m ¯ β = 0 and a = 0 . Then ( κ X i =0 x i : κ ∈ N , x , ..., x κ ∈ ∆ ) = ¯ X. Proof.
Notice that m ≥ deg( β ) + 1. We have T a m ◦ ... ◦ T a (0) = 037ence the set B : = { T a k ◦ ... ◦ T a (0) : 1 ≤ k ≤ m − } = (cid:8) a ¯ β k − + ... + a k − ¯ β + a k ¯ β : 1 ≤ k ≤ m − (cid:9) is a subset of ∆. Set∆(0) = ( κ X i =0 x i : κ ∈ N , x , ..., x κ ∈ ∆ ) . The proof is completed by showing inductively that ¯ β , ..., ¯ β m − ∈ ∆(0).Indeed ¯ β ∈ B ⊆ ∆ and if ¯ β , ..., ¯ β k ∈ ∆(0), for κ < m −
1, then¯ β k +1 = a (( a ¯ β k +1 + ... + a k +1 ¯ β + a k +2 ¯ β ) − a ¯ β k − ... − a k ¯ β − a k +2 ¯ β ) ∈ ∆(0) . In this section we will prove Lemma 4.2
Proof.
By bringing the matrix to it’s normal form of a reducible matrix, see([23], p. 51), we can assume that A = B ∗ ∗ · · · ∗ B ∗ · · · ∗ ... ... ... ...0 0 0 · · · ∗ · · · B h where B i is a non-negative irreducible square matrix for i ∈ { , ..., h } . Byrescaling we can assume that ρ ( A ) = 1. Clearly 1 = ρ ( A ) = max { ρ ( B ) , ..., ρ ( B h ) } so from assumption iii) we get ρ ( B ) = 1. We set S i := { j ∈ { , ..., N } : The entry (j,j) is contained in the B i -block } . i ∈ { , ...h } let V i := (cid:8) u ∈ R N : u ( j ) = 0 if j / ∈ S i (cid:9) and V i − := (cid:8) u ∈ R N : u ( j ) = 0 if j / ∈ ∪ i − κ =1 S κ (cid:9) . Define p i and p i − to be the orthogonal projections of R N to the subspaces V i and V i − respectively. Finally let B ′ i to be A where all entries outside the B i -block are replaced by 0 and B ′ i − to be A where all the entries of the form( i, j ) are replaced by zero if and only if j / ∈ ∪ i − κ =1 S κ .We will prove the lemma by proving inductively that p i ( e A n ) converges for i ∈ { , ..., h } . For i = 1 we have that p i ( e A n ) = p i ( e B ′ n ) so the statementis true since B is an irreducible aperiodic matrix of spectral radius one. Theaperiodicity comes from assumption i). Now we assume that i ∈ { , ..., h } and p i − ( e A n ) converges to some v ′ ∈ R N aiming to prove that p i ( e A n )converges.Case 1 ρ ( B i ) <
1: We define T i : R N → R N by T i ( x ) = xB ′ i + p i ( v ′ A )Since ρ ( B i ) < u ′ ∈ R N such that u ′ ( I − B ′ i ) = p i ( v ′ A ) so that T i ( x ) = ( x − u ′ ) B ′ i + u ′ . Now, from ρ ( B i ) < T ni ( x ) → u ′ for any x ∈ R N .Writing p i ( e A n ) = T ni (0) + p i ( e A n ) − T ni (0)we only need to prove that p i ( e A n ) − T ni (0) → p i ( e A n ) to u ′ . Let ε >
0. By the spectral radius formula there exists
C > || B ′ ni || ≤ C ( ρ ( B i ) + δ ) n where δ > ρ ( B ′ i ) + δ <
1. Also by p i − ( e A n ) → v ′ weget that there is κ such that | p i ( v ′ A ) − p i ( p i − ( e A n − ) A ) | < ε . Notice that p i ( e A κ +1 ) = p i ( e A κ ) B ′ i + p i ( p i − ( e A κ )) , κ ∈ { , ... } .
39y iterating the relation above and choosing n large enough we get | p i ( e A n ) − T ni (0) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X κ =1 (cid:0) p i ( p i − ( e A κ − ) A ) − p i ( v ′ A ) (cid:1) B ′ n − κi (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) κ − X κ =1 (cid:0) p i ( p i − ( e A κ − ) A ) − p i ( v ′ A ) (cid:1) B ′ n − κi (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + n X κ = κ || B n − κi || · ε ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) κ − X κ =1 (cid:0) p i ( v ′ A ) − p i ( p i − ( e A κ − ) A (cid:1) B ′ κ − − κi ! B ′ n − κ +1 i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + ε · C − ρ ( B i ) − δ Since xB ′ ni → x ∈ R N the above giveslim sup n →∞ | p ni ( e A n ) − T ni (0) | ≤ ε · C − ρ ( B i ) − δ but since ε was arbitrary we getlim n →∞ | p ni ( e A n ) − T ni (0) | = 0completing the inductive step in the case ρ ( B i ) < ρ ( B i ) = 1: Now let u ′ be a left eigenvector of 1 of B ′ i with all entriesin S i being positive. There exists such a u ′ from Perron–Frobenius theoremsince B i is a non-negative irreducible matrix. There are κ , m ∈ N and c > S i of p i ( p i − ( e A n ) A m ) − cu ′ are positive for all n > κ . This is true, by choosing c small enough, becauseof assumption ii) and p i − ( e A n ) → v ′ . Let κ ∈ N be such that m ( κ − >κ . The inequalities in the following are to be understood entrywise. For n p i ( e A nm ) = n X κ =1 (cid:0) p i (cid:0) p i − (cid:0) e A m ( κ − (cid:1) A m (cid:1)(cid:1) B ′ m ( n − κ ) i ≥ n X κ = κ (cid:0) p i (cid:0) p i − (cid:0) e A m ( κ − (cid:1) A m (cid:1)(cid:1) B ′ m ( n − κ ) i = n X κ = κ (cid:0) p i (cid:0) p i − (cid:0) e A m ( κ − (cid:1) A m (cid:1) − cu ′ (cid:1) B ′ m ( n − κ ) i + n X κ = κ cu ′ B ′ m ( n − κ ) i ≥ n X κ = κ cu ′ B ′ m ( n − κ ) i = ( n − κ + 1) cu ′ . The above implies that || p i ( e A nm ) || → ∞ which contradicts assumptioniii). Thus case 2 never occurs. We have a number of further questions on the structure of the sets X ( β ), themeasure µ , and on how one can start to study µ using ergodic theory. Question 1:
Is it the case for any integer alphabet A and for any hyperbolic β one can express X ( β ) (or the higher dimensional analogue ˜ X ( β ) in the non-Pisot case) as a cut and project set with window R (or maybe R o ) defined asthe attractor of an iterated function system { S i : i ∈ A} where S i is definedin terms of the Galois conjugates of β of absolute value less than one? Wehave shown an inclusion in Corollary 4.1. This question is also considered in[14]. Question 2:
Is it true that, for a sequence of Pisot numbers β n of increasingdegree in any interval (1 , − ǫ ), the sequence of sets β n − (cid:16) X {− , , } ( β n ) ∩ h − β − , β − i(cid:17) equidistribute in [ − , X {− , , } ( β n )renormalised to live on [ − , β n ,the distance between measures µ I βn and normalised Lebesgue measure on I β n n tends to infinity. Our question here is the correspondingquestion for the sets supp( µ I βn ) = X {− , , } ( β n ) ∩ h − β − , β − i . If the answerto Question 1 is positive, then this is a question about the structure of asequence of cut and project sets. Question 3:
Does further numerical evidence support our Conjectures 1and 2 on the dimension of Bernoulli convolutions and the distribution ofmeasures µ I βn ? The case that β n is a sequence of Pisot numbers convergingto a limit in (1 ,
2) is of particular interest. In that case the limit must alsobe a Pisot number.
Question 4:
In the special case of the Golden mean, Theorem 3.3 describeshow the measure µ evolves as one moves through the spectrum. Can one usethis theorem, for example, to prove that the sequence of probability measureslim n →∞ P x ∈ X ( φ ) ∩ [0 ,n ] µ { x } X x ∈ X ( φ ) ∩ [0 ,n ] µ { x } δ x ( mod converges weak ∗ to Lebesgue measure on [0 , { ( x, y, z ) : y ∈ [0 , φ ] } we have an irrational rotation in the x direction,and an irrational rotation in the y direction which also gives the weightswhich tell us how to evolve the measure µ . Then one might believe our ques-tion has a positive answer, since the weights µ ( x ) are driven by the evolutionin the y direction which is somehow independent of our position in the x direction. Tom Kempton is partially supported by EPSRC grant EP/T010835/1. Weare grateful to Paul Mercat, Nikita Sidorov and Tom´aˇs V´avra for usefuldiscussions.
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