Generalized de Bruijn digraphs and the distribution of patterns in -expansions

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1 Discrete Mathematics 263 (2003) wwwelseviercom/locate/disc Generalized de Bruijn digraphs and the distribution of patterns in -expansions Wolfgang Steiner Institut fur Geometrie, Technische Universitat Wien, Wiedner Hauptstrae 8-10=113, A-1040 Wien, Austria Received 29 June 2000; received in revised form 27 June 2001; accepted 10 December 2001 Abstract A generalization of de Bruijn digraphs is dened for Parry s -expansions and it is shown that the characteristic polynomial of these graphs is in principle that of With the help of this result we prove that certain functionals of -expansions, eg the number of specic digital patterns, satisfy a central limit theorem, which is an extension of a result due to Drmota (Proceedings of the International Conference held in Graz, Austria, 1998, p 103) c 2002 Elsevier Science BV All rights reserved Keywords: De Bruijn graphs; Characteristic polynomial; Digital expansions; Central limit theorem 1 Introduction Our starting points are the -expansion of real numbers due to Parry [14] and the induced linear recurrence (cf [12]) Let 1 be a real number Then the -expansion of an arbitrary real number x is given by x = ; where 1 =[x], the integer part of x, and the other digits are computed with the transformation T (x)={x} (where {x} denotes the fractional part of x): n =[T n 2 (x)] This work was supported by the Austrian Science Foundation FWF, grant S8302-MAT Fax: address: steiner@geometrietuwienacat (W Steiner) URL: X/03/$ - see front matter c 2002 Elsevier Science BV All rights reserved PII: S X(02)

2 248 W Steiner / Discrete Mathematics 263 (2003) Then, the digits j satisfy ( n ; n+1 ;:::) (a 1 ;a 2 ;:::) for n 2; where the a j s are the digits of the -expansion of, ie = a 1 + a 2 + a and denotes the lexicographic order In particular, we have (a n ;a n+1 ;:::) (a 1 ;a 2 ;:::) for n 2: (11) Conversely, if a sequence (a 1 ;a 2 ;:::) satises (11), we have a real number with -expansion (a 1 ;a 2 ;:::) Those which have recurrent tails in their -expansions, ie a j+m = a j for all j n for some integers n and m, are called -numbers The -numbers which have a nite -expansion are called simple -numbers A simple -number (a r 0, a j = 0 for all j r) is a root of the polynomial x r a 1 x r 1 a r 1 x a r ; which is called characteristic polynomial of If is a non-simple -number, it is a root of the polynomial (x n+m a 1 x n+m 1 a n+m 1 x a n+m ) (x n a 1 x n 1 a n 1 x a n ): This polynomial is called characteristic polynomial, if n and m are minimal with this property For the induced linear recurrence we distinguish between simple -numbers and other real numbers For simple -numbers we dene j a i G j i + 1 for j r; i=1 G 0 =1; G j = (12) r a i G j i for j r and for the others i=1 G 0 =1; G j = j a i G j i + 1 for j 0: (13) i=1 With G =(G n ), every non-negative integer n has a (unique) proper G-ary digital expansion n = j 0 j (n)g j

3 with integer digits j (n) 0 such that k j (n)g j G k+1 j=0 The digits j = j (n) satisfy W Steiner / Discrete Mathematics 263 (2003) for k 0: ( k ; k 1 ;:::) (a 1 ;a 2 ;:::) for k 0 (14) (cf [12]), where we have set j = 0 for all j 0 Conversely, a sequence ( 0 ; 1 ;:::)is the digital expansion of an integer n if (14) holds and j 0 only for a nite number of j 0 We will study functions depending on subblocks of these digital expansions Let B L = {( L 1 (n); L 2 (n);:::; 0 (n)) : n G L } be the set of blocks B {0; 1;:::;a 1 } L of length L which actually occur in G-ary digital expansions Let F : B L+1 R be any given function (for some L 0) with F(0; 0;:::;0) = 0 Furthermore, set s F (n)= j 0 F( j+l (n); j+l 1 (n);:::; j (n)): This means that we consider a weighted sum over all subsequent digital patterns of length L + 1 of the digital expansion of n For example, for L = 0 and F()= we just obtain the sum-of-digits function, or if L = 1 and F(; )=1 ; ( x;y denoting the Kronecker delta) then s F (n) is just counting the number of times that a digit is dierent from the preceding one, etc In order to get an insight into the distribution of s F (n), it is convenient to consider a related sequence of random variables X N, N 1, dened by Pr[X N 6x]= 1 N {n N: s F(n)6x} In Section 3 we will show that Drmota s methods in [3] can be applied to prove asymptotic normality of the distribution of X N Drmota showed this for the special case of nite recurrences of the type (12) with a 1 a 2 a r 0 It should be noted that the case L = 0 for general is treated by Drmota and Gajdosik [4] In the case of simple -numbers we will use a generalization of de Bruijn digraphs (Denition 21) and, in particular, the property that the characteristic polynomial of these graphs is the characteristic polynomial of multiplied with a factor x n This property has been conjectured by Drmota [3] and is shown in Section 2 Remark 11 Sometimes dierent generalizations of de Bruijn digraphs can be found in the literature Imase and Itoh [9] and Reddy et al [16] introduced independently the graph G B (n; d), where n d, the set of vertices is {0; 1;:::;n 1} and the set of edges consists of i di + r (mod n) for 06i n; 06r d:

4 250 W Steiner / Discrete Mathematics 263 (2003) Later, Imase and Itoh [10] introduced also the graph G l (n; d) whose set of vertices is the same as that of G B (n; d) and the set of edges consists of i d(n 1 i)+r (mod n) for 06i n; 06r d: Du and Hwang [5] showed that these graphs have some of the properties of de Bruijn digraphs Their characteristic polynomial has been given by Xueliang and Fuji [18] 2 Generalized de Bruijn digraphs For a q-ary expansion which is a special case of Parry s expansion with = q an integer, we have B L = {0; 1;:::;q 1} L Then B L is the set of vertices of a directed de Bruijn digraph with edges B =( 1 ;:::; L ) C =( 1 ;:::; L ) where ( 2 ;:::; L )=( 1 ;:::; L 1 ): (21) With this characterization of de Bruijn digraphs we can generalize them on Parry s -expansions with simple -numbers Denition 21 Let (a 1 ;a 2 ;:::;a r )bear-tuple which satises (11) (if we set a n := 0 for n r) and B L be the set of blocks ( 1 ;:::; L ) which satisfy ( k ; k+1 ;:::) (a 1 ;a 2 ;:::) for k 1 (if we set n := 0 for n L) Then the generalized de Bruijn digraph of B L is dened by the set of vertices B L and the edges B C where B =( 1 ;:::; L ) and C =( 1 ;:::; L ) satisfy (21) and ( 1 ;:::; L ; L ) B L+1 : (22) Remark 22 Because of (14), this denition of B L is equivalent to the one given in the Introduction Remark 23 If (21) holds and L r, then (22) is automatically satised For L = r 1, the only exceptions are ( 1 ;:::; r 1 )=(a 1 ;:::;a r 1 ); ( 1 ;:::; r 1 )=(a 2 ;:::;a r 1 ;x) with a r 6x6a 1 Remark 24 An important property of de Bruijn digraphs is that the line graph of the graph of blocks of length L is the graph of blocks of length L + 1 For our generalization, this property holds if we have L r 1 Denote by A L the adjacency matrix of the generalized de Bruijn digraph of B L

5 W Steiner / Discrete Mathematics 263 (2003) Example 25 For the q-ary expansion with q = 2 (or equivalently the G-ary expansion with r =1, a 1 = 2), we have B 2 = {(0; 0); (0; 1); (1; 0); (1; 1)} and A 2 = : Example 26 In the case of the Zeckendorf expansion (r =2, a 1 = a 2 = 1, the G j are the Fibonacci numbers) we have B 2 = {(0; 0); (0; 1); (1; 0)} and A 2 = : The characteristic polynomial of these graphs is strongly connected to the characteristic polynomial of the corresponding simple -number Theorem 27 For L r 1, the characteristic polynomial of the adjacency matrix A L of the generalized de Bruijn digraph of B L is (A L )(x)=x GL r p(x); where G =(G j ) j 0 is dened by the nite linear recurrence (12) and p(x)=x r a 1 x r 1 a 2 x r 2 a r 1 x a r is the characteristic polynomial of the linear recurrence and of the corresponding simple -number Proof First we remark that #(B L )=G L and for each B B L B =( L 1 (i 1); L 2 (i 1);:::; 0 (i 1)) for some i {1; 2;:::;G L } where the j s are the digits in the G-ary expansion Conditions (21) and (22) can thus be written for i; j {1; 2;:::;G L } as and ( L 2 (i 1);:::; 0 (i 1))=( L 1 (j 1);:::; 1 (j 1)) (23) ( L 1 (i 1); L 1 (j 1);:::; 0 (j 1)) B L+1 ; (24) respectively Therefore, the coecients of A L =(a (L) ij ) 16i; j6g L are { 1 if (23) and (24) hold; a (L) ij = 0 otherwise:

6 252 W Steiner / Discrete Mathematics 263 (2003) First assume L r Then, we can omit (24) (cf Remark 23) and have a (L) i+kg L 1;j = a(l) i; j : We dene the matrix P L := (p (L) ij ) 16i; j6gl for L r by 1 if i = j; p (L) ij := 1 if j6g L 1 ; i= j + kg L 1 ; k 0; 0 otherwise: Hence, P 1 L =(p ( L) ij ) 16i; j6gl has the coecients 1 if i = j; p ( L) ij = 1 if j6g L 1 ; i= j + kg L 1 ; k 0; 0 otherwise: With P L we dene a matrix similar to A L by A L =(a (L) ij ) 16i; j6gl := P L A L P 1 L : In the construction of A L the rows i of A L are subtracted from the rows i + kg L 1 and the columns i + kg L 1 are added to the columns i Therefore, a (L) ij = 0 for all i {G L 1 +1;G L 1 +2;:::;G L }; j {1; 2;:::;G L } and it suces to continue with the matrix ( ) 12 GL 1 A L 1 =(a (L 1) ij ) 16i; j6gl 1 := A L 12 G L 1 (this notation means that we take only the rows and columns 1; 2;:::;G L 1 of A L ) because of (A L )(x)=(a L)(x)=x GL GL 1 (A L 1 )(x): The so-dened A L 1 is the adjacency matrix of the generalized de Bruijn digraph of B L 1, because we have a (L 1) ij = 1 not only if (23) holds, but also for ( L 2 (i 1);:::; 0 (i 1))=( L 1 (j + kg L 1 1);:::; 1 (j + kg L 1 1)) =(k; L 2 (j 1);:::; 1 (j 1)) with a k {0; 1;:::;a 1 } (and j + kg L 1 6G L ), and therefore a (L 1) ij =[( L 3 (i 1);:::; 0 (i 1))=( L 2 (j 1);:::; 1 (j 1))];

7 W Steiner / Discrete Mathematics 263 (2003) where we use the Iversonian notation [expression]=1 if expression is true and 0 otherwise For L = r we have to check additionally a (r 1) ij = 0 for ( r 2 (i 1);:::; 0 (i 1))=(a 1 ;a 2 ;:::;a r 1 ); ( r 2 (j 1);:::; 0 (j 1))=(a 2 ;:::;a r 1 ;x) with a r 6x6a 1 (cf Remark 23) This is true, because for these i and j, a (r 1) would imply that we have a j = j + kg r 1 with a (r) ij = 1, ie ( r 1 (j 1);:::; 0 (j 1))=(a 1 ;a 2 ;:::;a r 1 ;x); but this violates (14) and is therefore impossible Hence, we iterate the construction until we get A r 1 and obtain ij =1 (A L )(x)=x GL Gr 1 (A r 1 )(x): (25) We continue with L = r 1 For i G r 1 ; j6g r 1 (but not for i = G r 1 ) we have and a (r 1) ij =[( r 3 (i 1);:::; 0 (i 1))=( r 2 (j 1);:::; 1 (j 1))] a (r 1) i+kg r 2;j = a(r 1) i; j : For 16s6r 1, we dene the matrix P s := (p (s) ij ) 16i; j6g r 1 by 1 if i = j; p (s) ij := 1 if j6g s 1 ; i= j + kg s 1 ; k 0 and i G s ; 0 otherwise: With A r 1 := P r 1A r 1 P 1 r 1,weget a (r 1) ij = 0 for all i {G r 2 +1;G r 2 +2;:::;G r 1 1};j {1; 2;:::;G r 1 } and build the matrix ( ) 1 2 Gr 2 G A r 2 := A r 1 r 1 ; 1 2 G r 2 G r 1 where the numeration of the rows and columns is kept, ie A r 2 = ) i; j {1;2;:::;Gr 2;G r 1} This matrix satises (a (r 2) ij (A r 1 )(x)=(a r 1)(x)=x Gr 1 Gr 2 1 (A r 2 )(x) and for i G r 2 ; j6g r 1 a (r 2) ij =[( r 4 (i 1);:::; 0 (i 1))=( r 3 (j 1);:::; 1 (j 1))] (but A r 2 is not the adjacency matrix of a generalized de Bruijn digraph)

8 254 W Steiner / Discrete Mathematics 263 (2003) Hence, we iterate this procedure by dening ( ) 1 2 Gs 1 G A s 1 := (P s A s Ps 1 s G r 1 ) 1 2 G s 1 G s G r 1 for 16s6r 2 and get, for i G s, j6g s a (s) ij =[( s 2 (i 1);:::; 0 (i 1))=( s 1 (j 1);:::; 1 (j 1))]: Therefore, we have (A r 1 )(x)=x Gr 1 r (A 0 )(x): (26) The denitions of P s and A s 1 imply for 16s6r 1, j6g s 1 and i {1; 2;:::;G s 1 ; G s ;:::;G r 1 }, and thus G s 1 a (s 1) ij = [ s (j ) s (j)][( s 2 (j );:::; 0 (j ))=( s 2 (j);:::; 0 (j))]a (s) ij j =1 G r 1 1 = = [ s (j ) s (j)][( s 2 (j );:::; 0 (j ))=( s 2 (j);:::; 0 (j))]a (r 1) ij a (0) ig s = a (s) ig s = j =1 16j G r 1: s( j) 1; ( s 1( j);:::; 0( j))=(0;0;:::;0) a (r 1) ij : (27) Now we can easily calculate A 0 =(a (0) ij ) i; j {G 0;G 1;:::;G r 1} For i = G t with 06t6r 1 we have ( r 2 (G t 1);:::; 0 (G t 1))=(0;:::;0; 0;a 1 ;a 2 ;:::;a t ): Therefore, we have a (r 1) G t;j = 1 if and only if ( r 2 (j 1);:::; 0 (j 1))=(0;:::;0;a 1 ;a 2 ;:::;a t ;x) (28) with 06x6a t+1 for t r 1 and if and only if ( r 2 (j 1);:::; 0 (j 1))=(a 2 ;a 3 ;:::;a r 1 ;x) (29) with 06x a r for t = r 1, respectively For the rst column, ie s = 0, the sum in (27) runs over all j with 0 (j) 1 x = a t+1 in (28) implies j = G t+1 and 0 (j) = 0, otherwise 0 (j)=x +1 1 For i = G t we have therefore a t+1 terms a (r 1) ij = 1 and a (0) G tg 0 = a t+1 For 16s6r 1 and i = G t, the conditions on j in (27) imply ( s 1 (j 1);:::; 0 (j 1))=(a 1 ;:::;a s )

9 W Steiner / Discrete Mathematics 263 (2003) and, with (28) and (29), we have a (r 1) G tj = 1 for at most one j Together with (11) we get x = a t+1 We have a (r 1) G r 1j = 0 for this j and hence a(0) G r 1G s = 0 For 06t r 1, the G-ary expansion of this j is ( t+1 (j);:::; 0 (j))=(1; 0; 0;:::;0): Thus, we obtain a (0) G tg s =[t +1=s] for s 0 Recapitulating, A 0 has the form a a A 0 = 0 1 a r 0 0 and its characteristic polynomial is clearly p(x) With (25) and (26), the theorem is proved Remark 28 For de Bruijn digraphs we have (A L )(x)=x ql 1 (x q) Remark 29 For general digraphs D, the characteristic polynomial of the line graph L(D) is a multiple of that of D: (L(D))(x)=(D)(x)x e(d) v(d) ; where e(d) denotes the number of edges and v(d) the number of vertices This property could be used in the rst part of the proof of Theorem 27 but anyway our construction is needed for the second part of the proof 3 Asymptotic properties of functions depending on subblocks of -expansions Now we study the random variables X N dened in the introduction Expected value and variance of X N are given by EX N = 1 s F (n) and by VX N = 1 N N n N We introduce the function (s F (n) EX N ) 2 : (31) n N c N (z)= n N sf (n) z

10 256 W Steiner / Discrete Mathematics 263 (2003) and consider for any block B =( 1 ;:::; L ) B L the functions aj B (z):= z sf (n) : n G j; ( j 1(n);:::; j L(n))=B Then B B L a B j (z)=c N (z): In order to obtain recurrent relations for the functions aj B we need the following notation: For B =( 1 ;:::; L ) B L let B =( 2 ;:::; L ) denote the block consisting of the last L 1 elements of B and B the rst element 1, ie B =( B ;B ) (Similarly B =( 1 ;:::; L 1 )) Furthermore, for (; B)=(; 1 ;:::; L ) B L+1 set L 1 (; B)= (F(0;:::;0;; 1 ;:::; L i ) F(0;:::;0; 0; 1 ;:::; L i )) i=0 Note that (0;B)=0 31 Simple -numbers + F(0;:::;0; 0;): In the case of simple -numbers, we may assume, without loss of generality, that L r 1 (If we are only interested in L + 1 subsequent digits with L r 1, then we consider a new function F : B r R that does not depend on the rst (r L 1) digits) Lemma 31 The functions aj B (z), j 0, are recursively given by aj B (z)= aj 1(z)z C (B;C) : C B L: C=B ;( B;C) B L+1 Proof The set {n G j : ( j 1 (n);:::; j L (n)) = B} is divided into subsets of the form {n G j : j 1 (n)= B ; ( j 2 (n); j 3 (n);:::; j L 1 (n))=(b ;)=C} = {n G j 1 : ( j 2 (n); j 3 (n);:::; j L 1 (n)) = C; ( B ;C) B L+1 } + B G j 1 : Because of {( B ;C): B; C B L ; C = B } B L+1 we cover all possible cases Furthermore, for n G j 1 with ( j 2 (n); j 3 (n);:::; j L 1 (n)) = C and ( B ;C) B L+1, we have s F (n + B G j 1 )=s F (n)+( B ;C):

11 W Steiner / Discrete Mathematics 263 (2003) Corollary 32 The vector a j (z)=(aj B (z)) B BL a j (z)=a L (z)a j 1 (z) (j 0); satises the matrix recursion where the G L G L -matrix A L (z)=(a B; C (z)) B; C BL is given by { z ( B;C) if C = B and ( B ;C) B L+1 ; a B; C (z)= 0 otherwise: A L (1) is the adjacency matrix of the generalized de Bruijn digraph of B L, its characteristic polynomial is therefore (Theorem 27) (A L (1))(x)=x GL r (x r a 1 x r 1 a 2 x r 2 a r 1 x a r ) and is an eigenvalue of A L (1) Lemma 33 shows that the other eigenvalues of A L (1) have absolute value less than Lemma 33 The conjugates of a simple -number with respect to the characteristic polynomial have absolute value less than Proof Set g(x):=1 x r p(x 1 )= r a j x j : j=1 If x, then g(x 1 ) 6g( x 1 ) g( 1 )=1 and p(x) 0 If x =, x, then we either have g(x 1 ) g( x 1 ) = 1 or all powers x j with a j 0 have the same argument, which must be dierent from 0 because of a 1 0 In both cases we have g(x 1 ) 1 and p(x) 0 Because of g ( 1 ) 0, is a simple root of p(x) and the lemma is proved Now we have all prerequesities for Drmota s proofs of the following lemma and theorems: Lemma 34 (cf Drmota [3, Lemma 33]) Let G(t; z) = det(ti A L (z)) be the characteristic polynomial of the matrix A L (z) Then there exists a (complex) neighbourhood of z =1 such that G(t; z)=0 has a unique solution t = (z) of maximal modulus Furthermore, the function (z) is analytic in this neighbourhood Theorem 35 (cf Drmota [3, Theorem 21]) and EX N = 1 N VX N = 1 N n N n N s F (n)= log N log + O(1) (s F (n) EX N ) 2 = 2 log N log + O(1);

12 258 W Steiner / Discrete Mathematics 263 (2003) where = (1) and 2 = (1) + 2 : Theorem 36 (cf Drmota [3, Theorem 22]) If 2 0, then for every 0 1 N {n N: s F(n) EX N + xvx N } = 1 x e (1=2)t2 dt + O((log N) 1=2+ ) 2 uniformly for all real x as N Theorem 37 (cf Drmota [3, Theorem 22]) If 2 0, F just attains integer values and d = gcd{(; B): (; B) B L+1 } =1; then for every 0 {n N: s F (n)=k} = N 2VXN (exp ( (k EX N ) 2 uniformly for all non-negative integers k as N 32 Non-simple -numbers 2VX N Now we treat the case of non-simple -numbers, ie (a n+m+1 ;a n+m+2 ;:::)=(a n+1 ;a n+2 ;:::) ) ) + O((log N) 1=2+ ) for some integers m; n and m; n minimal with this property Let F be a function F : B l+1 R as above We may assume that l = km n + m 1 for some k N (If we are interested in l + 1 subsequent digits with l km then we consider a new function F : B km+1 R that does not depend on the rst (km l) digits) Lemma 38 For all l N we can nd an integer L l such that (a j ;a j+1 ;:::;a L+1 ) (a 1 ;a 2 ;:::;a L j+2 ) for all j {2;:::;L+1} (32) Proof If (32) holds for L := l, we are nished Otherwise, we have a j6l + 1 such that (a j ;a j+1 ;:::;a l+1 )=(a 1 ;a 2 ;:::;a l j+2 ) and an integer g l + 1 such that (a j ;a j+1 ;:::;a g 1 )=(a 1 ;a 2 ;:::;a g j ) and a g a g j+1 : If (32) holds for L := g 1, we are nished Otherwise, we have a j 6g such that (a j ;a j +1;:::;a g )=(a 1 ;a 2 ;:::;a g j +1):

13 W Steiner / Discrete Mathematics 263 (2003) For j j, wehad (a j ;a j +1;:::;a g ) (a j j+1;a j j+2;:::;a g j+1 )6(a 1 ;a 2 ;:::;a g j +1): Therefore, j j We can nd g g such that (a j ;a j +1;:::;a g 1)=(a 1 ;a 2 ;:::;a g j ); a g a g j +1 and repeat this procedure Since j j j 1, we nd a L that satises (32) after a nite number of steps Remark 39 If (32) holds for L, it holds for L + m since (a j ;:::;a L+m+1 )=(a j m ;:::;a L+1 ) for all j {L +2;:::;L+ m +1} and, by induction, for L + km Now we consider the functions aj B (z):= n G j;( j 1(n);:::; j L(n))=B z sf (n) (B B L ): Lemma 310 Let L satisfy (32) and L km n+m 1 Then the functions aj B, j L, B B L are recursively given by a B j (z)= C B L: C=B ; ( B;C) B L+1 if B (a 1 ;a 2 ;:::;a L ) and a (a1;:::;al) j (z)= where and C=(a 2;:::;a L; L); L6a L+1 D C km a C j 1(z) z (B;C) C aj 1(z) z (a1;c) b D j L 1(z) z (a1;:::;al+1;d) ; (33) C km := {D B km : D (a L+2 ;a L+3 ;:::;a L+km+1 )}; bj D sf (i) (z):= z i G j: ( j 1(i);:::; j km (i))=d; ( j 1(i);:::; 0(i)) (a L+2;:::;a L+j+1) ( 1 ; 2 ;:::; L+1 ;D):=s F (n 1 ) s F (n 2 )

14 260 W Steiner / Discrete Mathematics 263 (2003) with D =( 1 ; 2 ;:::; km ); i (n 1 )= i (n 2 )=0 for all i km + L +1; ( km+l (n 1 ); km+l 1 (n 1 );:::; 0 (n 1 ))=( 1 ; 2 ;:::; L+1 ; 1 ; 2 ;:::; km ); ( km+l (n 2 ); km+l 1 (n 2 );:::; 0 (n 2 ))=(0;:::;0; 1 ; 2 ;:::; km ): The functions bj D (z), j km, D C km are recursively given by aj B (z) if D (a L+2 ;:::;a L+km+1 ); bj D B B L: ( 1;:::; km )=D (z)= bj km(z)z E (0;:::;0;aL+2;:::;aL+km+1;E) if D =(a L+2 ;:::;a L+km+1 ): E C km Proof The proof of the rst equation is the same as that of Lemma 31 We just have to check ( B ; j 2 (n);:::; 0 (n); 0;:::) (a 1 ;a 2 ;:::) (34) for C =( j 2 (n);:::; j L 1 (n)); C = B ; ( B ;C) B L+1 (34) can only be violated, if ( B ;C)=(a 1 ;:::;a L+1 ) which implies B =(a 1 ;:::;a L ) We have a (a1;:::;al) j (z)= ( j 1;:::; 0): ( j 1;:::; j L)=(a 1;:::;a L); j L 16a L+1; ( j 2;:::; 0) B j 1 ( j 1;:::; 0): ( j 1;:::; j L 1)=(a 1;:::;a L+1); ( j 2;:::; 0) B j 1; ( j L 2;:::; 0) (a L+2;:::;a j) sf ( j 1;:::; 0) z z sf ( j 1;:::; 0) ; (35) where s F ( j 1 ;:::; 0 ):=s F (n) for the (unique) integer n with G-ary expansion ( j 1 ; :::; 0 ) (and i = 0 for all i j) The rst sum of (35) is equal to the rst sum of (33) (cf Lemma 31 and consider (32)) The patterns of the second sum of (35) are exactly those of the rst sum which do not satisfy ( j 1 ;:::; 0 ) B j Because of (32) the choice of patterns ( j L 2 ;:::; 0 ) in the second sum of (35) is not inuenced by the patterns ( j 1 ;:::; j L 1 ) Therefore this sum is equal to the second sum of (33) The equation for bj D (z); D (a L+2 ;:::;a L+km+1 ) is clear For D = (a L+2 ;:::;a L+km+1 ) we have to consider Remark 39 and that (a L+km+2 ;:::;a L+2km+1 )= (a L+2 ;:::;a L+km+1 ) Corollary 311 The vector a j (z)=(a B1 j (z);:::;a BG L j (z);b D1 j 1 (z);:::;bdm j 1 (z);:::;bd1 j L (z);:::;bdm j L (z))t

15 W Steiner / Discrete Mathematics 263 (2003) with B i := ( L 1 (i 1);:::; 0 (i 1)); M := #(C km ); D 1 := (a L+2 ;:::;a L+km+1 );:::;D M := (a 1 ;:::;a km ) satises the matrix recursion a j (z)=a L (z)a j 1 (z) (j L); where A L (z)=(a i; j (z)) 16i; j6gl+lm is given by z (B i ;Bj) if i; j6g L ; B j = B i; ( Bi ;B j ) B L+1 ; z (a1;:::;al+1;d j G L (L 1)M ) if i = G L ; j G L +(L 1)M; z (0;:::;0;aL+2;:::;a L+km+1;D j GL (km 1)M ) if i = G L +1; j6g L + kmm; a i; j (z)= j G L +(km 1)M; 1 if G L +26i6G L + M; D i GL =( 1 ;:::; km )(B j ) or i G L + M; j = i M; 0 otherwise: Hence, if L km, A L (1) has the form à L E M E M 0 where à L is the matrix A L of the generalized de Bruijn digraph of the (L + 1)-tuple (a 1 ;a 2 ;:::;a L ;a L+1 + 1) and E M is the identity matrix of size M ;

16 262 W Steiner / Discrete Mathematics 263 (2003) A km (1) has the form 0 0 Ã km : E M E M 0 Theorem 312 The characteristic polynomial of A L (1) is where (A L (1))(x)=p(x) (x (k 1)m + x (k 2)m + +1)x GL+LM km n ; (36) p(x)=(x n+m a 1 x n+m 1 a n+m ) (x n a 1 x n 1 a n ) is the characteristic polynomial of Proof First we construct a matrix A =(a ij) 16i; j6gl+l with (A L (1))(x)=(A )(x) x LM L : To get A, we dene P h =(p (h) ij ) 16i; j6g L+LM ; 06h L; with p (h) i; i := 1 for all i6g L + LM; p (h) G L+hM+1;j := 1 for all j with G L + hm j6g L +(h +1)M: Then ( ) 12 GL +1 G A L + M +1 G L +(L 1)M +1 := A ; 12 G L +1 G L + M +1 G L +(L 1)M +1

17 W Steiner / Discrete Mathematics 263 (2003) where A := P 0 P 1 :::P L 1 A L P 1 L 1 P 1 L :::P 1 0 : A has the form 0 à L E L 1 0 if L km and 0 à km E L 1 0 if L = km, respectively Since à L is the matrix A L of the generalized de Bruijn digraph of the (L + 1)-tuple (a 1 ;a 2 ;:::;a L ;a L+1 + 1), it can be transformed to a a a L 1 a L (see the proof of Theorem 27) In the transformation, the last row is never added to or subtracted from another row Because of this and the fact that a ij= 0 for all i G L ; j G L ; we can apply this transformation to the whole matrix A and get the

18 264 W Steiner / Discrete Mathematics 263 (2003) (2L +1) (2L + 1)-matrix A which has the form a a a L a L y 0 y 1 y l and a a a L a L ; y 0 y 1 y l respectively, (A )(x)=(a )(x) x GL L 1 and the y j ; 06j6L; are given by y j := #(D j ) with D j = {B =( 1 ;:::; L ) B L : ( 1 ;:::; km ) (a L+2 ;:::;a L+km+1 ); ( L j+1 ;:::; L )=(a 1 ;:::;a j ); ( L+1 h ;:::; L ) (a 1 ;:::;a h ) for all h j}:

19 W Steiner / Discrete Mathematics 263 (2003) Lemma 313 The y L j ; 06j6L, are recursively given by y L =1; y L j = j a L+j+1 if 16j km; y L j+h a h a L+1+km +1 if j = km; 0 if j km: h=1 Proof y L = 1 is obvious If B =( 1 ;:::; L ) D L j, let h 1 be maximal with the property then B =( 1 ;:::; j h ;a 1 ;:::;a h 1 ; j ;a 1 ;:::;a L j ) ( 1 ;:::; j h ;a 1 ;:::;a L j+h ) D L j+h : If we take a L-tuple ( 1 ;:::; j h ;a 1 ;:::;a L j+h ) D L j+h,16h j, then ( 1 ;:::; L ):=( 1 ;:::; j h ;a 1 ;:::;a h 1 ; j ;a 1 ;:::;a L j ) D l j if and only if j a h and ( 1 ;:::; km ) (a L+2 ;:::;a L+km+1 ) If j km, the denition of ( 1 ;:::; L ) guarantees that the last condition holds If j6km, it provides ( 1 ;:::; j 1 ) (a L+2 ;:::;a L+j ): Therefore, ( 1 ;:::; km ) (a L+2 ;:::;a L+km+1 ) is violated if and only if j6km; ( 1 ;:::; j 1 )=(a L+2 ;:::;a L+j ); j a L+j+1 or j = km; ( 1 ;:::; j 1 )=(a L+2 ;:::;a L+j ); km 6a L+km+1 : We calculate (A ) by expanding det(xi A ) at the columns (2L + 1) and (L + km + 1): (A )(x)=x L km (x km (x L+1 a 1 x L a L x a L+1 1) +( 1) km ( 1) L+km+1+L+2 (x L+1 a 1 x L a L x a L+1 1)) +( 1) L 1 ( 1) 2L+1+L+1 det(â L ) = x L km (x km 1)(x L+1 a 1 x L a L x a L+1 1) det(â L ) = x L km (x (k 1)m + x (k 2)m + + 1)(x m 1) (x L+1 a 1 x L a L x a L+1 1) det(â L );

20 266 W Steiner / Discrete Mathematics 263 (2003) where  j ; 16j6L; is the matrix x a a 2 x 1  j = 0 0 ; 0 a j 0 0 x 1 y 0 y 1 y j Hence Therefore, det(â L )= y L (x L a 1 x L 1 a L ) + det(â L 1 )= = y L x L +(a 1 y L y L 1 )x L 1 +(a 2 y L + a 1 y L 1 y L 2 )x L 2 + +(a L y L + + a 1 y 1 y 0 ) = x L + a L+2 x L a L+km x L km+1 +(a L+km+1 +1)x L km = x L km (x km a L+2 x km 1 a L+km x a L+km+1 1) = x L km (x (k 1)m + + 1)(x m a L+2 x m 1 a L+m+1 1): (A )(x)=x L km (x (k 1)m + + 1)((x L+m+1 a 1 x L+m a L+m+1 1) (x L+1 a 1 x L a L+1 1)) = x 2L+1 km n (x (k 1)m + x (k 2)m + +1)p(x): (A L (1))(x)=x GL L 1+LM L (A )(x) and the theorem is proved = x GL+LM km n (x (k 1)m + x (k 2)m + +1)p(x) Hence is an eigenvalue of A L (1) and the other eigenvalues are 0, the roots of (x (k 1)m + x (k 2)m + + 1) which are kmth roots of unity and the roots of the characteristic polynomial of As for simple -numbers, is the eigenvalue with the largest absolute value, which is shown in Lemma 314 Lemma 314 The conjugates of a non-simple -number with respect to the characteristic polynomial have absolute value less than

21 Proof Set, for x 1, f(x):=1 a 1 x a 2 x 2 a 3 x 3 and, for x 1, g(x):=1 f(x 1 )= W Steiner / Discrete Mathematics 263 (2003) a j x j : j=1 Then, for the same reasons as in the proof of Lemma 33, the roots of f(x) have absolute value less than for x If we set then p k (x):=p(x)(1 + x m + x 2m + + x (k 1)m ) p k (x)=(x n+km a 1 x n+km 1 a n+km ) (x n a 1 x n 1 a n ): With q k (x):=x n km p k (x) we have q k (x)= ( 1 a 1 x a ) n+km x n+km ( 1 x km a 1 a ) n xkm+1 x n+km and, for x 1, q k (x) f(x) ask Therefore f(x) = 0 is a necessary condition for p(x)=0, x 1, and the roots of p(x) have absolute value less than for x Since q k () 0 for some suciently big k, is a simple root of p(x) and the lemma is proved Hence Lemma 34, Theorems 35 and 36 are also valid for non-simple -numbers, whereas the proof of Theorem 37 cannot be directly applied since it uses properties of non-negative matrices and A L (1) contains negative elements in this case References [1] G Barat, RF Tichy, R Tijdeman, Digital blocks in linear numeration systems, in: Number Theory in Progress, Vol 2, Zakopane-Koscielisko, 1997, de Gruyter, Berlin, 1999, pp [2] NL Bassily, I Katai, Distribution of the values of q-additive functions on polynomial sequences, Acta Math Hungar 68 (1995) [3] M Drmota, The distribution of patterns in digital expansions, in: F Halter-Koch, RF Tichy (Eds), Algebraic Number Theory and Diophantine Analysis, Proceedings of the International Conference held in Graz, Austria, 1998, de Gruyter, Berlin, 2000, pp [4] M Drmota, J Gajdosik, The distribution of the sum-of-digits function, J Theorie Nombres Bordeaux 10 (1998) [5] DZ Du, FK Hwang, Generalized de Bruijn digraphs, Networks 18 (1) (1988) [6] JM Dumont, A Thomas, Gaussian asymptotic properties of the sum-of-digits function, J Number Theory 62 (1997) 19 38

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