Guy Barat and Peter J. Grabner

Transcription

1 DISTRIBUTION PROPERTIES OF G-ADDITIVE FUNCTIONS Guy Barat and Peter J Grabner Abstract We extend previous results of Delange [De] concerning the existence of distribution functions of certain q-adic digital functions to the case of digital expansions with respect to linear recurrences with decreasing coefficients Furthermore we investigate a special case of digital functions and give a functional equation for the related distribution function We prove uniqueness and continuity of the solution of this equation Introduction Let us recall that if G n is an increasing sequence of positive integers with n G 0 =, we can expand every positive integer with respect to this sequence, ie n N, n = ε k G k, this expansion being finite and unique, provided that for every K, K ε kg k < G K The digits ε k can be computed by the greedy algorithm In this paper, we will only study the case that the base sequence is a recurrence sequence with decreasing integer coefficients a 0 a a d, namely : 2 G n+d = a 0 G n+d + +a d G n for n 0 with G 0 = and G k = a 0 G k + +a k G 0 + for k < d The initial values are chosen as the canonical initial values from [GT] In this case, a finite sum ε kg k is the expansion of some integer iff 3 ε k,,ε 0,0 < a 0,,a d < being understood as the lexicographical order for every k The ε k,,ε 0 that verify this condition are called admissible strings The dominating root of the characteristic equation X d a 0 X d a d = 0, say α, is a Pisot number; G n these equations have been studied by Brauer [Br] In particular, lim n α exists n and is non-zero For detailed discussion of these numbers systems, we refer to [Fr,GT,GLT] The first author is partially supported by URA-CNRS Nr 225 The second author is supported by the Schrödinger scholarship J00936-PHY Typeset by AMS-TEX

2 2 GUY BARAT AND PETER J GRABNER We want to study the notion of G-additive functions which generalize A O Gelfond s [Ge] q-additive functions and we will give a sufficient condition for the existence of a related distribution function More precisely, we consider arithmetic functions which satisfy K K 4 f ε k G k = fε k G k for which we investigate existence and properties of 5 Fx := lim N N n<n χ x fn =: lim N F Nx where χ x denotes the characteristic function of the interval ],x[ This is a continuation of investigations initiated by Delange [De], who gave a necessary and sufficient condition for the existence of a distribution function in the q-adic case These investigations complement considerations concerning uniform distribution of G-additive and also q-additive functions modulo cf [MF], [GT] We will give a sufficient condition for the existence of such a function in the Brauer case For this purpose we will use a result of Kooman [Ko,Ko2], who derived an extension of the theory of Poincaré and Perron concerning linear recurrences with non-constant coefficients We will give some indications, why it seems to be difficult to give also necessary conditions In a last part we study in more details a special case of G-additive functions, which satisfy 6 f 0ε 0 ε K 0 = βf ε 0 ε K 0 where K ε kg k is the G-expansion of any integer and β being smaller than We give a functional equation satisfied by the distribution function The ideas which led to the functional equation also give a proof of the continuity of F for every non-trivial function f of this type We shall use the convention that = 0 Double brackets will indicate intervals of N 2 Existence of the distribution function According to Delange, we say that f admits a distribution function F if the limit 5 exists for every point of continuity of F By basic Fourier-Stieltjes transform theory see for example [Ch], we know that a necessary and sufficient condition for that to hold is the pointwise convergence of the characteristic functions Φ N t = e itx df N x = N n<n e itx δ fn x = N n<n e itfn to a function Φ continuous at 0 In fact, an almost everywhere convergence is enough, provided that Φ0 =, but we will not use this refinement Moreover, if that situation holds, then Φ is the characteristic function of F We first state three lemmata

3 DISTRIBUTION PROPERTIES OF G-ADDITIVE FUNCTIONS 3 Lemma Let m be a positive integer, and x k 0 k<m a finite sequence of real numbers Then, the following inequality holds: m m 2 e itx k m t x k + t2 m x 2 k 2 Proof m m e itx k m it x k = m m txk 0 t 2 x 2 k 2 e iu du because e iu u for every u The lemma follows immediately The second lemma is a special case of a theorem due to Kooman [Ko,Ko2] Lemma 2 Suppose we have a linear recurrence with non-constant coefficients Z n+d = a n 0 Z n+d + +a n d Z n Suppose further that there exist a k for k = 0,,d, a d 0 such that 22 n a k a k n=0 converges for every k = 0,,d and that P := X d a 0 X d a d has distinct roots α i Then initial values Z i 0,,Zi d for i =,,d can be chosen such that the corresponding solutions Z n,,z n d satisfy lim n Z i n α n i for i =,,d In particular, if there is one dominating root α, then the limit Z n lim n α n exists for every solution of the recurrence The third one is a result which is part of the folklore in the study of digital functions Special cases are proved in [CRT] and [GLT] Lemma 3 Let g be a G-multiplicative function of modulus less or equal to, where G is a basis given by the recurrence 2 Assume further that exists, then exists also lim k lim N = gn G k n<g k N gn n<n

4 4 GUY BARAT AND PETER J GRABNER Theorem 4 Let f be a G-additive function, such that the series a s 23 fa 0 G n+d + +a s G n+d s +lg n+d s n=0 converges for every s = 0,,d and such that the series 24 a 0 flg n 2 n=0 converges, then the distribution function exists and its Fourier-Stieltjes transform is equal to the limit for N of Proof Let Ξ N be defined as : Φ N t = N e itfn n<n Ξ N t := G N Φ GN t For an arbitrary integer n, let us consider the G-expansion of any integer m smaller than G n+d : m = n+d ε k G k Because of the admissibility of the string ε n+d,,ε n, the G-expansion of m is of one of the following types : a 0 G n+d +a G n+d 2 + +a d 2 G n+ +kg n +r, with k a d and r < G n a 0 G n+d +a G n+d 2 + +a d 3 G n+2 +kg n+ +r, 25 with k a d 2 and r < G n+ a 0 G n+d +kg n+d 2 +r with k a and r < G n+d 2 kg n+d +r with k a 0 and r < G n+d Conversely, all those sums represent a unique integer smaller than G n+d Thus, we have the following recurrence formula for Ξ : 26 Ξ n+d t = a0 e itfkg n+d e itfa 0G n+d a e itf d 2 j=0 a jg n+d j Ξ n+d t+ e itfkg n+d 2 ad e itfkg n Ξ n+d 2 t+ + Ξ n t

5 with for s = 0,,d DISTRIBUTION PROPERTIES OF G-ADDITIVE FUNCTIONS 5 Ξ s t = First step : We prove that Ξ nt α n n<g s e itfn has a limit for every t, where we recall that α is the dominating root For that, we show that the relation 24 verifies the assumptions of Lemma 2 Fixing t, we have Ξ n+d t = a n 0 tξ n+d t + + a n d tξ nt, where, for any s, 0 s < d, a n s t = e itf s j=0 a jg n+d j as e itfkg n+d s Using Lemma, we get: a s s a n s t a s t fa j G n+d j +fkg n+d s j=0 + t2 a s s fa j G n+d j +fkg n+d s 2 j=0 The convergence of the series a s s fa j G n+d j +fkg n+d s n=0 j=0 is exactly condition 23 Furthermore we have s fa j G n+d j +fkg n+d s j=0 fa j G n+d j 2 +2s 2 fkg n+d s 2 s 2s j=0 by Schwarz inequality Hence, a s s fa j G n+d j +fkg n+d s 2sa s s j=0 j=0 a s fa j G n+d j 2 +2s 2 2 fkg n+d s 2 d 2 d 2d a 0 fa j G n+d j 2 +2d 2 j=0 s=0 a fkg n+d s 2

6 6 GUY BARAT AND PETER J GRABNER Thus, a s n=0 s fa j G n+d j +fkg n+d s j=0 2d 2 a 0 +d a 0 flg n 2 n=0 We are now allowed to apply Lemma 2, which tells us that Ξ nt α n has a limit for every t Second step : We use the result of the first step to prove that the limit of Θ n t := Ξ nt continuous function of t Θ n t satisfies Θ n+d t = an 0 t α Θ n+d t+ an t Θ α 2 n+d 2 t + an 2 d t α d α n Θ n t, and the conditions of Lemma 2 remain satisfied for this recurrence Let K be a compact subset of R For every s and every n, the function t a n s t a s is continuous on K, and has a maximum on it that we note m n s Let M n n be the sequence defined by M n+d = a 0 +m n 0 M n+d + + a d +m n d M n with M k = sup Ξ k t for 0 k d t K By a trivial recurrence on n, we have Ξ n t M n for every n N and every t K Moreover, Ξ n+d t Ξ n+d t a n 0 t Ξ n+d t + a n t Ξ n+d 2 t + is a + n a d t Ξn t a 0 +m n 0 M n+d + a +m n M n+d + + a d +m n d M n M n+d M n+d We can apply Lemma 2 to the sequence M n n, every series n=0 mn s being equal to a convergent series n=0 an s t s a s Thus, M n n is convergent, and the series Ξ n+ t Ξ n t is normally convergent Hence, its limit is continuous n=0 Third step : Up to now, we have proved that G nφ Gn t α n admits a continuous limit But lim n G n α n exists and is non-zero, so Φ G n t n admits a continuous limit

7 DISTRIBUTION PROPERTIES OF G-ADDITIVE FUNCTIONS 7 Now, f being G-additive, e itf is G-multiplicative, and we can apply Lemma 3 to finish the proof of Theorem 4 Remark : It seems to be difficult to obtain a necessary condition for the distribution function to exist The limitation comes from the perturbation part of the proof, namely Kooman s Lemma 2 His result does not hold if absolute convergence of the coefficients is not assumed, but the Z n G n could converge, even if the series n an k a k diverge, as the following counterexample shows : With Z 0 = G 0 and Z = G Z n+2 = a Z n+ + a 2 + G n+ Z n n ng n = a Z n+ +a 2 Z n In the following we will use ideas from [GLT] and [Ma] to give an alternative approach to the existence of a distribution function in a more specific case: According to [GLT], the G-compactification of N is defined as K G := {ε n n N ; k N, ε k,,ε 0 is admissible}, where K G is endowed with the topology induced by that of the infinite product of discrete spaces + n=0 [[0;a 0]] The addition of can be extended on K G, where it is a continuous map With this operator, it has been shown in the same paper that the so-called adding machine is uniquely ergodic We will note P the corresponding probability measure, and consider K G as a Borel probability space We identify N and its image in K G If C is the cylinder [ε 0,,ε K ], then C has probability 27 F K α d +F K+ a 0 F K α d 2 + +F K+d a 0 F K+d 2 a d 2 F K, α K α d +α d where F K := # { n < G K ; n C } The difference between this formula and that of [GLT] is due to a modification of the notations and to a printing error in the previous paper Proposition 5 Let f be a G-additive function Then f can be extended to a continuous function on K G if and only if 28 a 0 flg n < n=0 Then the distribution function of the extension is exactly F except, maybe, at the points of discontinuity of F Proof We first prove the equivalence of the existence of a continuation to K G with 28: Suppose first that 28 holds Then the series n=0 fε ng n is absolutely convergent for any infinite string ε n n N in K G Thus, f can be extended on the compactum to f, whose continuity is a trivial consequence of 28 and of the product topology structure

8 8 GUY BARAT AND PETER J GRABNER Conversely, suppose that a 0 flg n = n=0 Then there is an l 0 [[0,a 0 ]] for which fl 0 G n = n=0 Then there is a k [[0,d ]] such that the series 29 fl 0 G k+dn n=0 diverges Define now { l0 if fl 0 G k+nd > 0 b n = 0 else { l0 if fl 0 G k+nd < 0 c n = 0 else The strings 0 k b 0 0 d b 0 d b 2 and 0 k c 0 0 d c 0 d c 2 are admissible, since 0 k l 0 0 d is admissible By the divergence of 29 at least one of those strings corresponds to an infinite value of the extension Since the adding machine is uniquely ergodic, every point, in particular 0, is generic Thus, for any Riemann-integrable complex-valued function g on K G, we have 20 N N n=0 gn N gdp K G In particular, if we take g to be χ t f, and if we note F the distribution function of the integrable function f, 20 yields Ft = Ft Of course, the previous argument is valid if χ t f is Riemann-integrable, ie iff F is continuous at the point t Remarks : Proposition 5 gives a new and extremely short proof for the existence of the distribution function, provided that 28 holds Indeed, 28 is stronger than 23, and this probabilistic approach does not help in the most general case Consider, eg, G n+2 = 3G n+ +3G n and the G-additive function defined by fg n = n+, f2g n = n+ and f3g n = 0 Then the series 23 and 24 converge, whereas 28 diverges This is also a strong difference between Mauclaire s study of arithmetic additive functions [Ma] and the G-additive functions we deal with 3 Application to a particular case Recall that the shift adjoint operator associated to the basis G n n is defined as follows: S : N N ε k G k ε k G k+ k k

9 DISTRIBUTION PROPERTIES OF G-ADDITIVE FUNCTIONS 9 In this part, we study a special kind of G-additive functions, which satisfy 3 n N : fsn = βfn where β < this is clearly a rewriting of 6 A typical example of such a function is f ε k G k := ε k β k Anyway, f is given by its values on [[;a 0 ]] as it yields from f ε k G k = β k fε k Proposition 6 If αβ <, then F has zero derivative almost everywhere, in the sense of Lebesgue measure µ The Hausdorff dimension of the set where the derivative does not vanish is at most Proof Clearly, fn K=0 ε K,,ε 0 admissible logα log β [ K f ε k G k β K fε K K ; f ε k G k + β K fε K ] Moreover, the set of admissible ε K,,ε 0 has cardinal G K+, and if we define λ to be max 0 ε a 0 fε, then µ ε K,,ε 0 admissible [ K f ε k G k β K fε K K ; f ε k G k + ] β K fε K is smaller than or equal to 2λ β K G K+ and hence belongs to O αβ K Suppose now αβ < fn is an enumerable intersection of sets whose measure tends to zero Hence, µ fn = 0 This shows that the derivative of F is 0 on the complement of a set of measure 0 The statement concerning Hausdorff dimension is an immediate consequence of the fact that fn can be covered by G k intervals of length λβ k for all positive integers k For suitable choice of f at the digits there is equality, but it is also possible that the dimension breaks down We now try to obtain a functional equation for F For this, we exhibit a tiling of [[0;G K [[ involving the shift operator, to allow us to use 3 Let K be an integer, and n [[0;G K+ [[ Writing n = K ε kg k, we define ln := min{k [[0;K ]]; ε k < a d } ; If the previous set is empty, ln is defined as Thus, n can be uniquely written as follows : 32 n = S l+ m+r with 0 l K, m < and r P l { K } or n Q K := ε k G k ; ε k a d for each k

10 0 GUY BARAT AND PETER J GRABNER where { } l P l = m N; m = ε k G k, k < l : ε k a d, 0 ε l < a d We are now able to write F GK in a different way G K+ F GK+ t = n<g K+ χ t fn If β > 0, we get = K 33 G K+ F GK+ t = If β < 0, we get χ t m< K f S l+ m+x + m=0 y Q K χ t fy χt fx fm+ y Q K χ t fy 34 G K+ F GK+ t = K l [2] + K l 0[2] m=0 m=0 χt fx fm+ y Q K χ t fy χt fx fm For convenience, we will henceforth consider positive β We will mention the results obtained for negative ones, the computations being quite identical We state now some intermediate results Lemma 7 Let us define p l := #P l Then the generating function of the p l is given by Pz = In particular p l z l +z + +z d = a d a 0 a d z a d 2 a d z d p l is convergent to α αl Proof By 32, we have the following recurrence formula : G K+ = K p l + p K+ a d Translating this recurrence into a relation of the generating functions Pz and Gz of p n and G n resp yields Gz = zgzpz+ a d Pz or Pz = Gz zgz+ a d

11 DISTRIBUTION PROPERTIES OF G-ADDITIVE FUNCTIONS Gz is given by cf [GT] Gz = Thus we have the desired result for Pz and +z + +z d a 0 z a d z d α = P = α n=0 p n α n Remark It is an immediate consequence of this lemma and the definition of Hausdorff dimension that the dimension of f l P l is at most for β α, where γ is the dominating root of the polynomial logγ log β z d a 0 a d z d 2 a a d z d 3 a d 2 a d Remark Notice that in the case a 0 = a = = a d = a the P l are only non-empty for l = 0,,d Furthermore, in the case a = a 2 = = a d = a, a 0 = a+ the number of elements of the P l is ad for l d These are the only cases where p l remains bounded Corollary 8 The following series converges normally : F α l+ t fx Proof α l+ F t fx p l α l+, whose sum converges by the previous Lemma 7 We denote H the function defined by Corollary 8 Dividing 33 by G K+ and subtracting Ht we get F GK+ t Ht = K α l+ F + G K+ K G K+ t fx y Q K χ t fy l=k+ m=0 α l+ F χt fx fm t fx

12 2 GUY BARAT AND PETER J GRABNER Thus, for any integer S smaller than K we have F GK+ t Ht l=s+ S G K+ K G K+ S α l+ K l=s+ α l+ α l+ m=0 m=0 t fx F α l+ l=k+ S K G K+ α l+ p l +C S α l+ +2 K l=s+ p l α + l+ l=k+ m=0 l=s+ α l+ m=0 m=0 χt fx fm χt fx fm F χt fx fm F + G K+ p l α l+ y Q K χ t fy χt fx fm F p l α l+ + p K+ a d G K+ t fx χt fx fm t fx t fx where C is a constant such that G K+ C α l+ The existence of C comes from this of two constants A and B verifying Aα n G n Bα n for every n Let ε > 0 There exists S in N such that max2, C l=s+ p l < ε αl+ Now, there exists an integer K 0 greater than S such that for every K > K 0 the following majorations hold : S G K+ α l+ p l < ε l < S, x P l : t fx t fx F F εαl+ S +p l p K+ < ε a d G K+ Then, for any K > K 0, we have FGK+ H 7ε Thus we have proved the following :

13 DISTRIBUTION PROPERTIES OF G-ADDITIVE FUNCTIONS 3 Theorem 9 i/ For positive β, F is a solution of the functional equation 35 Yt = α l+ Y t fx ii/ For negative β, F is a solution of the functional equation 36 Yt = l α l+ Y t fx + p 2l α 2l+ In order to be able to use the above functional equations to determine the distribution functions we prove the following: Theorem 0 The solutions of the functional equations 35 and 36 are unique among distribution functions, where it is understood that two distribution functions yielding the same measure are identified Proof We first prove that a solution of the functional equation 35 has to have compact support in the sense that 0 < f < is bounded Since the function fx is bounded we can choose t 0 such that t < t fx for all t > t 0 and for all l and x P l Then we have for such t and any solution F of 35 Ft = F α l+ t fx α l+p lft = Ft; the last equality is a consequence of Lemma 7 Therefore we have t fx 37 Ft = F for all l Take now l =, x fixed and t large enough; then F is constant on the interval [t, t fx β ] Take t 2 2 = t fx β > t 2 β and continue this geometrically increasing process, we derive that the function is constant on the interval [t, [; this constant has to be The argument also applies for large negative values of t For negative β one has to split summation in 36 into odd and even l; the same arguments as above yield Ft+F t = for large enough t Then again 37 has to hold, and thus F has compact support To prove uniqueness we compute the moment generating function of 35 and 36 ie ˆFz = e tz dft we have omitted the i in the usual Fourier-Stieltjes transform just for convenience The transform of any solution is an entire function because of the property proved above It satisfies 38 ˆFz = l= α l ˆFβ l zg l z,

14 4 GUY BARAT AND PETER J GRABNER where G l z = e zfx = n=0 z n n! fx n Since ˆF is an entire function we can write its power series expansion ˆFz = n=0 z n ϕn, ϕ0 = n! and insert into 36 which yields 39 ϕn βn β n n α P n β k l = ϕk fx n k, for n α k α where Pz is the generating function of the number of elements of the P s whose properties have been studied in Lemma 7 39 is a recurrence relation for the ϕn, thus all of them are uniquely determined by this equation and therefore the distribution function is uniquely determined in any point of continuity Proposition 5 gives the existence of a continuous extension of f to K G We will use the properties of this extension in order to prove the continuity of F Lemma Let s N Then the following event is almost sure wrt P: A s := { ω = ε 0,,ε k, K G ; l,,l s N s, j [[0;s ]],ε l ++l j +j,,ε l ++l j+ +j P lj+ } l= Proof A C s = { ω K G ; #{j N;ε j a d } < s } yields to P s A C s = P [x x m ] PA C 0 m=0l =0 l m =0 x P l x m P l m It is hence sufficient to prove that PA 0 = But PA 0 = because F l++j = G j for any x P l, 27 gives P [x] and, P [x] = G 0α d +G a 0 G 0 α d 2 + +G d a 0 G d 2 a d 2 G 0 α l+ +α+ +α d Thus, by Lemma 7, PA 0 = p l = αl+

15 DISTRIBUTION PROPERTIES OF G-ADDITIVE FUNCTIONS 5 Theorem 2 Let f be of the 3 type, and F its distribution function Then F is continuous iff f is not identically 0 Proof If f is 0, then F is the Heaviside function χ [0, [, which is not continuous at t = 0 In general, if f satisfies 3, then f verifies the assumptions of Proposition 5 too In terms of P, the continuity of F means that Pf = t = 0 for every t we henceforth omit the tilde Let us note π l to be the l-th projection on K G Then f is given by fω = f π lωβ l Let t R Then, considering ω K G, taking an s in N and conditioning wrt the almost sure event A s, we have P fω = t = P fω = t A s = = = l =0 l =0 l =0 l s =0 l s =0 P fω = t 0 j < s,ε l + +l j +j,,ε l + +l j+ +j P lj+ Pπ 0 ω,,π l ++l s +s ω = x,,x s x P l x s P l s P fω = t π 0 ω,,π l ++l s +s ω = x,,x s l s =0 x P l P x s P l s P π 0 ω,,π l + +l s +s ω = x,,x s fω = t fx x s β l + +l s +s We apply now the previous equality to some t for which P fω = t attains its greatest value Because of P π 0 ω,,π l + +l s ω = x,,x s =, l =0 l s =0 x P l x s P l s we have P fω = t fx x s = P fω = t for any x = x x s where β l + +l s +s x j belongs to P lj It is sufficient to remark that t fx/β l ++l s +s takes infinitely many values provided that f is not identically 0 to get a contradiction if P fω = t > 0 Indeed, suppose there exists some ε < a d such that fε 0 Considering x = 0 l ε, t/ fε/β takes infinitely many values if t 0 But for t = 0, we get a non zero value, namely fε/β, at which f reaches its maximum, and to which we are allowed to apply the argument above On the other side, if such an ε does not exist, there is an η a d such that fη 0 Considering x = η0 l, we apply the same argument mutatis mutandis, which ends the proof 4 Concluding Remarks Finally we want to present two special examples which show the applicability of the functional equations 35 and 36 We will show the uniform distribution of two van der Corput type digital sequences for the definition of van der Corput sequence we refer to [KN]

16 6 GUY BARAT AND PETER J GRABNER Proposition 3 Let a and d be positive integers and define G n by the recurrence G n+d = ag n+d ++G n and the corresponding canonical initial values Then the G-adic van der Corput sequence K f ε k G k = K ε k α k+ is uniformly distributed in [0,] The discrepancy of fn ie D N = sup x [0,] N χ x fn x n<n is O logn N Proof The proof is done just by inserting the distribution function of uniform distribution into 35 Since the solution is unique by Theorem 0, we are done The proof of the estimate for the discrepancy is a rephrasing of the proof for the q-adic van der Corput sequence cf [KN] Proposition 4 Let a be a positive integer and define G n by the recurrence G n+2 = a+g n+ +ag n and the initial values G 0 = and G = a+2 Define a G-additive function by fε = ε a for 0 ε a and fa+ = α α and K f ε k G k = Then fn is uniformly distributed in [0,] K fε k α k Proof The proof is again done by inserting the distribution function of uniform distribution into 35 and observing that is a disjoint union [fx,fx+ α = [0, l The two examples we presented here, are the only uniformly distributed G- additive functions we could find In the general case the functional equation expresses some self-similarity structure of the graph of the distribution function which shall be illustrated by the following picture We have computed the distribution function for G n+3 = 3G n+2 +G n+ +G n with initial values G 0 =, G = 4 and G 2 = 4 and K f ε k G k = K ε k α k+

17 DISTRIBUTION PROPERTIES OF G-ADDITIVE FUNCTIONS 7 The deviation from uniform distribution comes from the fact that the reversion of an admissible string in G-adic expansion need not be an admissible string in α-expansion cf [Pa] Acknowledgements This work was done during the second authors visit at the Dynamics, Statistics and Algorithms group of the Université de Provence The authors want to thank Pierre Liardet and Robert Tichy for many helpful

18 8 GUY BARAT AND PETER J GRABNER discussions The first author is indebted to Enrique Andjel for the idea which led to the proof of Theorem 2 References [Br] A Brauer, On algebraic equations with all but one root in the interior of the unit circle, Math Nachr 4 95, [Ch] K Chandrasekharan, Classical Fourier transforms, 980, Universitex, Springer Verlag [CRT] J Coquet, G Rhin et Ph Toffin, Représentation des entiers naturels et indépendence statistique 2, Ann Inst Fourier 3 98, 5 [De] H Delange, Sur les fonctions q-additives et q-multiplicatives, Acta Arith 2 972, [Fr] AS Fraenkel, Systems of numeration, Amer Math Monthly , 05 4 [Ge] AO Gelfond, Sur les nombres qui ont des propriétés additives et multiplicatives données, Acta Arith 3 968, [GLT] PJ Grabner, P Liardet and RF Tichy, Odometers and Systems of Numeration, Acta Arith , [GT] PJ Grabner and RF Tichy, Contributions to Digit Expansions with Respect to Linear Recurring Sequences, J Number Th , [Ko] RJ Kooman, Convergence Properties of Recurrence Sequences, 989, Thesis, University of Leiden [Ko2] RJ Kooman, Decomposition of matrix sequences, Indag Math new series 5 994, 6 79 [KN] L Kuipers and H Niederreiter, Uniform Distribution of Sequences, J Wiley and Sons, New York, 974 [Ma] J-L Mauclaire, Intégration et théorie des nombres, Hermann, Paris, 986 [MF] M Mendès-France, Les suites à spectre vide, J Number Th 5 973, 5 [Pa] W Parry, On the β-expansion of Real Numbers, Acta Math Acad Sci Hung 2 96, G Barat DSA Université de Provence CMI, Château Gombert 39, rue Joliot-Curie 3453 Marseille, Cedex 3 France P Grabner Institut für Mathematik A Technische Universität Graz Steyrergasse Graz Austria address: barat@ariochuniv-mrsfr grabner@weylmathtu-grazacat