NONNEGATIVE RADIX REPRESENTATIONS FOR. THE ORTHANT R n + JEFFREY C. LAGARIAS AND YANG WANG

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1 TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Vlume 348, Number 1, January 1996 NONNEGATIVE RADIX REPRESENTATIONS FOR THE ORTHANT R n + JEFFREY C. LAGARIAS AND YANG WANG Abstract. Let A be a nnnegative real matrix which is expanding, i.e. with all eigenvalues jj > 1, and suppse that j det(a)j is an integer. Let D cnsist f exactly j det(a)j nnnegativevectrs in R n.we classify all pairs (A D) such that every x in the rthant R n + has at least ne radix expansin in base A using digits in D. The matrix A must be a diagnal matrix times a permutatin matrix. In additin A must be similar t an integer matrix, but need nt be an integer matrix. In all cases the digit set D can be diagnally scaled t lie in Z n. The prfs generalize a methd f Odlyzk, previusly used t classify the ne{dimensinal case. 1. Intrductin Fr radix expansins t base b, the standard digit set D = f0 1 ::: b; 1g has the prperty thatevery real number x has at least ne radix expansin f the frm (1.1) x = 1X j=;k d j b ;j k2 Z and all d j 2D: We call digit sets with this prperty feasible fr base b, fllwing Odlyzk [12]. Knuth [8] raised the questin f describing, fr base 10, all feasible digit sets D f size 10. There are indeed feasible digit sets fr base 10 ther than the standard ne, fr example, the nn{standard digit set D = f g. Mre generally, ne can ask the same questin fr an arbitrary integer base b with jbj2, see Matula [11]. An imprtant bject in studying feasibility f a digit set is the set n 1X (1.2) T (A D) = A ;j d j : all d j 2D j=0 which is a cmpact set that satises the [ set{valued functinal equatin (1.3) A(T )= (T + d): If (b D) is feasible then (1.1) gives R = 1[ d2d j=0 b j T (b D) Received by the editrs July 1, Mathematics Subject Classicatin. Primary 11A63 Secndary 05B45, 39B42. Research supprted in part by the Natinal Science Fundatin, grant DMS{ c1996 American Mathematical Sciety

2 100 J. C. LAGARIAS AND Y. WANG and this shws that T (b D) must have psitive Lebesgue measure. In terms f T a necessary and sucient cnditin fr feasibility f a pair (b D) is that T cntains an pen interval which has 0 in its clsure. This criterin is nt easy t check, hwever. It is easy t see that if jdj < jbj then D cannt be feasible, fr (1.3) implies that the Lebesgue measure f T (b D) is 0. On the ther hand, when jdj > jbj there are many feasible digit sets, and the task f classifying them seems intractable. The mst interesting case ccurs fr feasible digit sets with jdj = jbj, whichwe call minimal feasible. In this case the representatins f real numbers (1.1) using a minimal feasible digit set D are essentially irredundant. Mre precisely, fr a feasible digit set each real x has nly nitely many expansins (1.1), and aside frm a set f Lebesgue measure zer, each x has a cnstant number f representatins, this number being 1 r 2, depending n D. If 0 lies in the interir f T (A D) this number is 2, and it is 1 therwise. Hwever even the prblem f classifying all the minimal feasible digit sets appears dicult, and it currently remains an pen prblem. In 1978 Odlyzk gave a cmplete classicatin fr the special case f nnnegative minimal feasible pairs (b D). Let R + := fx : x 0g and Z + := R + \ Z. Odlyzk [12] prved the fllwing: Therem 1.1 (Odlyzk 1978). Let b 2 Z + with b 2. Suppse that E i, 0 i k, are subsets f f0 1 ::: b; 1g such that (1.4) (1.5) (1.6) Then fr any >0 the digit set (1.7) E 0 + E E k = f0 1 ::: b; 1g je 0 jje 1 jje k j = b 1 2E 0 : D = E 0 + be b k E k is minimal feasible fr base b. Cnversely, if D is minimal feasible fr base b and cnsists f b nnnegative elements, then D is f the frm (1.7) fr sme >0 and sme subsets E 0 E 1 ::: E k f f0 1 ::: b; 1g satisfying (1.4){(1.6). Odlyzk's prf shwed that T (b D) is then a nite unin f intervals f length. In [9]we bserved that fr b 2 a cnverse result hlds, that if jdj = b and T (b D) is a nite unin f intervals, then fr sme translate D 0 = D + x the pair (b D 0 ) is nnnegative feasible. An imprtant feature f Odlyzk's prf is that it reduces the classicatin prblem t a prblem f factring cycltmic plynmials int zer{ne plynmial factrs. All pssible zer{ne factrizatins were determined earlier by Carlitz and Mser [3]. This paper frmulates and prves an n{dimensinal generalizatin f Therem 1.1. Let A 2 M n (R) be an expanding matrix and D R n. We saythe digit set D is feasible fr base A, r simply (A D) is feasible, ifevery x 2 R n can be represented in the frm (1.8) x = Q 1 X j=;k A ;j d j k2 Z and all d j 2D where Q is f the frm Q = diag(1 1 ::: 1). We saythat(a D) is nnnegative if A is a nnnegative matrix and D cnsists f nnnegative vectrs.

3 NONNEGATIVE RADIX REPRESENTATIONS FOR THE ORTHANT R n As in the ne{dimensinal case, assciated t any pair(a D) is the cmpact set n 1X (1.9) T (A D) = A ;j d j : all d j 2D : j=0 It satises the set{valued functinal equatin [ (1.10) A(T )= (T + d) d2d and is the attractr f the iterated functin system f' i : 1 i jdjggiven by ' i (x) =A ;1 (x + d i ) d i 2D: The feasibility cnditin (1.8) implies that (1.11) R n = [ Q=diag(1 ::: 1) Q 1 [ j=0 A j; T (A D) which shws that T = T (A D) has psitive Lebesgue measure. Nw (1.10) implies that there are n feasible sets D with jdj < j det(a)j. Wesay that D is minimal feasible if jdj = j det(a)j thisisthecasewe cnsider in this paper. When (A D) is minimal feasible then T is a self{ane tile in the terminlgy f Lagarias and Wang [9], [10]. Our bject in this paper is t classify thse nnnegative (A D) inr n such that D is a nnnegative minimal feasible digit set. The verall structure f the prfs fllw Odlyzk's apprach. Hwever sme new phenmena appear in dimensins n 2, and there are necessarily extra cmplicatins in the prfs. A key feature f the prfs is a determinatin f the structure f the assciated tile T (A D). Mre precisely, let R n + := f[x 1 ::: x n ] T : all x i 0g dente the nnnegative rthant inr n and let Z n + := Rn + \ Zn. It is clear that a nnnegative pair(a D) is feasible if and nly if all x 2 R n + have at least ne radix expansin (1.12) x = 1X j=;k A ;j d j k2 Z and all d j 2D: Our rst main result asserts that, unlike the ne{dimensinal case, there are substantial restrictins n the nnnegative matrices A that pssess a nnnegative minimal feasible digit set. We shw: Therem 1.2. Let A 2 M n (R) be a nnnegative and expanding matrix with j det(a)j = b that has a nnnegative feasible digit set with jdj = b. Then (1.13) A = BP in which B is a psitive diagnal matrix and P is a permutatin matrix. Furthermre ifb = diag(b 1 ::: b n ) and the permutatin assciated tp has cyclic decmpsitin = 1 2 k then (1.14) b j := Y i2 j b i 2 Z fr 1 j k with all b j feasible digit set D. 2. Cnversely, fr every such A there exists a nnnegative minimal

4 102 J. C. LAGARIAS AND Y. WANG Therem 1.2 is derived as a cnsequence f the strnger Therem 5.2 prved in Sectin 5. The matrix A in (1.13) is always expanding, but B need nt always be expanding, see Example 5.1 in Sectin 5. In Sectin 2 we establish the necessary cnditin A = BP in Therem 1.2, which implies that A k must be a diagnal matrix fr sme k 1. T prceed we study the special case f diagnal matrices B and btain: Therem 1.3. Let B = diag(b 1 ::: b n ) be an expanding diagnal matrix, and let DR n + be a nnnegative minimal feasible digit set fr base B. Then there exist nnnegative ne{dimensinal digit sets D 1 ::: D n R n +,each D i is minimal feasible fr base b i, such that (1.15) In particular, (1.16) D = D 1 D 2 D n : T (A D) =T (b 1 D 1 ) T (b n D n ): This result cmbines with Odlyzk's classicatin f ne{dimensinal nnnegative digit sets t give a cmplete classicatin fr nnnegative diagnal matrices. We establish Therem 1.3 in several steps. First, in Sectin 3 we shw that with a suitable scale change in D we reduce t the case that DZ n, and, mre imprtantly, then shw that T (A D) is a nite unin f lattice n{cubes (Therem 3.2). Once this is dne, the prblem is transfrmed t questins cncerning factrizatins f multivariate plynmials with zer{ne cecients int factrs f a similar frm. Fr this we prve a multivariate generalizatin f the criterin f Odlyzk (Therem 4.1). Finally, in Sectin 5 we cnsider the case f general nnnegative A f the frm (1.13). Using the fact that if (A D) is a nnnegative minimal feasible digit set, we bserve insectin2thatsis(a k D A k ) where D A k := X nk;1 i=0 A j d j : each d j 2D We maychse A k diagnal and then Therem 1.3 applies t (A k D A k ). We explit this fact t classify general (A D) in Therem 5.2. At the same time we deduce the necessary cnditin (1.14) and cmplete the prf f Therem 1.2 as a crllary. The assumptin f nnnegativity is crucial t all the results f this paper. Withut this restrictin there are minimal feasible digit set whse assciated regin T (A D) has a fractal bundary, r where T (A D) cnsists f innitely many cnnected cmpnents. Sme examples can be fund in Barnsley [2], Gilbert [4] and Vince [13]. Fr minimal feasible digit sets the regin T (A D) tiles R n,such tiles are studied in [1], [4], [5], [7], [9], [10], [13]. We thank A. M. Odlyzk fr helpful discussins. 2. Nnnegative Feasible Pairs: General Prperties In this sectin we assume the feasibility f(a D), s that jdj j det(a)j. Iterating the functinal equatin (1.10) yields [ (2.1) A m (T )= (T + d) d2d A m :

5 NONNEGATIVE RADIX REPRESENTATIONS FOR THE ORTHANT R n where (2.2) D A m := nm;1 X j=0 Then fr arbitrary digit sets we have: A j d j : each d j 2D : Lemma 2.1. Fr any m 1, the pair (A D) is feasible if and nly if the pair (A m D A m ) is feasible. Prf. The lemma fllws directly frm n 1 X j=;k A ;j d j : k 2 Z all d j 2D = n 1 X j=;k A ;mj d j : k 2 Z all d j 2D A m : Next, we assume nnnegativity and shw: Lemma 2.2. Suppse that (A D) is nnnegative and feasible. Then 0 2D. Prf. If 0 62 Dthen 0 cannt have a radix expansin, i.e., fr all k 2 Z and d j 2D 0 6= 1X j=;k This cntradicts the feasibility f(a D). A ;j d j : The rthant{cvering prperty (1.12) puts a signicant restrictin n the pssible frm f A, which frms the necessary cnditin (1.13) in Therem 1.3. Lemma 2.3. Suppse that (A D) is nnnegative and feasible. Then A = BP where B is a nnnegative diagnal matrix and P is a permutatin matrix. In particular A m is a diagnal matrix fr sme m>0. Prf. Since (A D) is nnnegative and feasible, R n + = n 1X j=;k A ;j d j : k 2 Z all d j 2D S A(R n + )=Rn + and A must map the x i{axis t sme x j {axis fr each 1 i n. Hence A = BP fr sme diagnal B and permutatin matrix P. If is a permutatin, its assciated permutatin matrix P has and we let act n diagnal matrices by Then fr any diagnal matrix B (P ) i (i) =1 (B) i i = B (i) (i) : BP = P ; ;1 (B) : Using this we cnclude that (BP ) k = B 0 P k fr sme diagnal matrix B 0, which implies that A n! is a diagnal matrix. :

6 104 J. C. LAGARIAS AND Y. WANG Nte that in Lemma 2.3 A must always be an expanding matrix fr feasible digit sets t exist, but, the matrix B in Lemma 2.3 is nt necessarily expanding, see Example 5.1 in Sectin 5. Lemma 2.1 and Lemma 2.3 reduce the study f nnnegative feasible pairs (A D) t thse fr diagnal matrices, except that we must reslve which digit sets n A m are f the frm D A m fr sme digit set D n A. We accmplish this in Sectin 5 fr nnnegative minimal feasible digit sets, after rst classifying the allwable frm f D A m when A m is diagnal in Sectin 4. Next, we suppse that (A D) is a nnnegative minimal feasible pair, and we deduce sme facts cncerning T = T (A D). The relatin (1.12) implies 1[ R n + = A m; (2.3) T (A D) m=0 s T has psitive Lebesgue measure (T ) > 0. Nw D A m has cardinality (2.4) jd A m j jdj m = j det(a)j m and taking the Lebesgue measure f bth sides f (2.1), using (T ) > 0implies that jd A m j = j det(a)j m and the measure{disjintness prperty (2.5) (2.6) ; (T + d 1 ) \ (T + d 2 ) =0ifd 1 d 2 2D A m, d 1 6= d 2. Lemma 2.2 nw shws that 0 2D,swehave Nw set (2.7) D = D A 1 D A 2 D A 3 : D A 1 = 1[ m=1 D A m : In view f (2.6) the measure{disjintness prperty extends t (2.8) ; (T + d 1 ) \ (T + d 2 ) = 0 fr d 1 d 2 2D A 1, d 1 6= d 2. Furthermre the relatin (2.3) nw becmes [ (2.9) R n + = (T + d): d2d A 1 By measure{disjintness this says that the rthant R n + is perfectly tiled with cpies f T, centered at pints f D A 1. (This is actually a self{replicating tiling f R n in the sense f Kenyn [6], [7].) Fr this reasn we call T (A D) the tile assciated t D. 3. Diagnal Case: Structure f Tile Suppse that B is a nnnegative diagnal matrix that is expanding. There is then cnsiderable freedm t rescale the digit set D. Fr any psitive real factrs s 1 s 2 ::: s n set S =diag(s 1 s 2 ::: s n ) and dene (3.1) S(D) := n d 0 = Sd =[s 1 d 1 s 2 d 2 ::: s n d n ] T : d =[d 1 d 2 ::: d n ] T 2D If (B D) is feasible, then s is (B S(D)) and vice versa, since (3.2) T ; B S(D) = S ; T (B D) : This equality is a cnsequence f (1.9) because S cmmutes with B.

7 NONNEGATIVE RADIX REPRESENTATIONS FOR THE ORTHANT R n Lemma 3.1. Let B = diag(b 1 b 2 ::: b n ) and suppse that (B D) is nnnegative and minimal feasible. Fr any subset I f1 2 ::: ng let D I dente the subset f the digit set D cnsisting f all digits d whse j{th crdinate d j =0fr all j 62 I. Then (3.3) jd I j = Y i2i b i : In particular all b i are integers. If B I := diag(b i : i 2 I) and D I is re{interpreted as a set f vectrs in R jij by drpping crdinates utside I, then(b I D I ) is minimal feasible in R jij. Prf. We use the radix expansin (1.12) cnned t the I{face f R n +,whichis (3.4) R I + n[x := 1 x 2 ::: x n ] T : all x j 0 and x j =0ifj 62 I : Since B is diagnal, all x 2 R I + can be represented by a radix expansin (1.12) using digits in D I.Thus S k0 Bk I ; T (BI D I ) cvers R jij +, hence (B I D I ) is feasible in R I.AlsT (B I D I ) has psitive jij{dimensinal Lebesgue measure, which frces jd I jdet(b I )= Y i2i b i : T shw that equality ccurs, we cnsider all jd I j k representatives D I B k = X nk;1 j=0 B j d j : all d j 2D I : Then fr any f =[f 1 ::: f n ] T 2D I B k,itsi{th crdinate f i fr i 2 I satises jf i j = X k;1 b j i d j i <Cb k i j=0 in which Q Q C = max d2d jdj 1 is a cnstant. S there can be at mst i2i (Cbk i )= C jij ( i2i Q b i) k dierent elements in DB k I.Bychsing k suciently large we btain jd I j i2i b i. This prves (3.4). The fact that b i 2 Z fllws by chsing I = fig and, the feasibility f(b I D I ) fllws frm the feasibility f(b D). Therem 3.2. Suppse that B = diag(b 1 ::: b n ) with all b i > 1, andthatd is a nnnegative digit set. Then (B D) is minimal feasible if and nly if jdj = j det(b)j and there exist psitive scaling factrs s 1 s 2 ::: s n with S = diag(s 1 s 2 ::: s n ) such that the scaled digit set D = S(D) has the fllwing tw prperties: (i) D Z n, and furthermre D cntains all 2 n zer{ne vectrs. (A zer{ne vectr is a vectr whse entries are all0 r 1:) (ii) T (B D ) is a nite unin f lattice cubes, i.e. there exists a nite set EZ n + cntaining 0 such that (3.5) T (B D )= [ f2e(f +[0 1] n ): Mrever, ; E;E \ ; D B 1 ;D B 1 = f0g. Fr any such (B D) the scaling factrs S fr which (i), (ii) hld are unique.

8 106 J. C. LAGARIAS AND Y. WANG Prf. First we shw that prperties (i), (ii) are sucient. [0 1] n T (B D ). Nw (3.2) gives whence 1[ [0 s ;1 1 ] [0 s;1 n ] T (B D) R n + B m; [0 s ;1 1 ] [0 s;1 n ] m=1 m=1 1[ Accrding t (ii), B m; T (B D) : This shws that D is feasible, and it is minimal by hypthesis. Cnversely, suppse (B D) is minimal feasible. We prve the existence and uniqueness f S such that (i), (ii) hld by inductin n the dimensin n. The base case n = 1 was established by Odlyzk ([12], Lemma 5, and his equatin (3.3)). The assertin ; E;E \ ; D B 1 ;D B 1 = f0g is a cnsequence f the measure{disjintness prperty (2.8). Suppse the therem is true fr dimensins up t n ; 1. We cnsider the sets I i = f1 2 ::: ngnfig and apply the inductin hypthesis n each fthe (n ; 1){dimensinal rthants R Ii + bunding R n +,t(b Ii D Ii ), which wemayd by Lemma 3.1. In each case we get unique scaling factrs (s (i) 1 ::: ^s(i) i ::: s (i) n ) where ^s (i) i is mitted, and these rescale the attractrs T (B Ii D Ii )sthateachisa nite unin f disjint (n ; 1){dimensinal unit cubes. Furthermre they must be cnsistent with each ther where they are bth dened, i.e., s (i) j = s (k) j whenever i k 6= j. Swehave a unique set f scaling factrs (s 1 s 2 ::: s n ), s i = s (k) i fr any k 6= i, such thatd = S(D) cntains 0 and all zer{ne vectrs except pssibly e 1 + e e n =[1 1 ::: 1] T : Als, the inductin hypthesis implies that any digit d 2D cntaining a zer entry necessarily lies in Z n.we d nt yet knw that D Z n,hwever. We prceed by a series f claims. Claim 1. T = T (B D )cntains the unit cube [0 1] n. Suppse nt. If all nnzer digits d 2D have sme crdinate at least 1, then all jb k dj 1 1 since B is diagnal. Thus the nly expansin (1.12) having x 2 (0 1) n wuld have k<0, s [0 1] n T. Hence there must be sme digit z =[z 1 ::: z n ] T 2D with 0 <z i < 1 fr all i. (If sme z i = 0 then we knw z 2 Z n.) Fr each 1 i n we dente w i = n min f i : f =[f 1 ::: f n ] : T f2d nz n S 0 <w i < 1fralli. This implies that fr any 0 6= d 2D,allB k d with k 0 either give vectrs utside the unit cube r else have the i{th crdinate at least w i,hencet must cntain the slab s S i =[0 1] i;1 [0 w i ] [0 1] n;i n[ S i T: i=1

9 NONNEGATIVE RADIX REPRESENTATIONS FOR THE ORTHANT R n Nw e 2 2D and we cmpare T +e 2 with T +z where z 2D with all its crdinates 0 <z i < 1. We get a cntradictin by shwing that (3.6) (T + z) \ (T + e 2 ) > 0 cntradicting the measure{disjintness prperty (2.8).T shw(3.6)we need nly t bserve that T + e 2 S 1 + e 2 =[0 1] [1 1+w 2 ] [0 1] n;2 T + z S 2 + z =[z 1 w 1 + z 1 ] [z 2 z 2 +1][z n z n +1]: Since 0 <z i w i < 1 fr all i, bth T + z and T + e 2 cntain the small cube [z 1 1 ::: 1] T + "[0 1] n fr sme suciently small ", which establishes (3.6). Thus Claim 1 fllws. Claim 2. e := e 1 + e e n =[1 1 ::: 1] T 2D. Suppse nt. Nw, [0 1] n T,sB(T [ ) [0 2] n. Using the self{ane prperty (3.7) B(T )= (T + d) d2d and the prperty thatt + d and T are measure{disjint fr all d 2D,theremust be a small cube (3.8) e + "[0 1] n T: T see this, nte that at least ne T +d cvers e+"[0 1] n when " is suciently small. Fr this t happen, d =[d 1 ::: d n ] n must satisfy d i 1+" fr all i. Nwnnef the translated tiles T + d, 0 6= d 2D cntains any interir pint fe + "[0 1] n,fr if they did then T wuld verlap sme d + T,cntradicting measure{disjintness. Hence (3.8) hlds. Nw we btain a cntradictin using the self{ane prperty (3.7). The tile T cntains [0 1] n,s@t must cntain all (n ; 1){dimensinal faces f the unit cube [0 1] n, fr if nt T wuld verlap at least ne translated tile T +d fr sme zer{ne digit 0 6= d 2D. S the bundary f B(T )cntains all (n ; 1){dimensinal faces f [0 b 1 ] [0 b n ]. But the S tile T has the unit cube with at least a small cube e + "[0 1] n attached t it, and d2d(t + d) can never prduce the \upper" at face [0 b 1 ] [0 b n;1 ] fb n g, because the prjecting piece ruins it. Thus B(T ) cannt be tiled by translates f T, a cntradictin that prves Claim 2. Claim 3. T is a nite unin f lattice cubes and D Z n. We assign a ttal rdering t Z n + = Z n \R n + with the prperty that if jgj 1 < jg 0 j 1 then g g 0 in the rdering, where jxj 1 := P n i=1 jx ij fr any x 2 R n. (There are many such rderings, and all we need is ne f them.) We prve the fllwing hypthesis by inductin n g 2 Z n +: (i) ; T \ (g +[0 1) n ) > 0 implies g +[0 1) n T. (ii) D \ (g +[0 1) n ) Z n. Fr g = 0, the smallest element inz n + with respect t the rdering, the hypthesis is clearly true: we have[0 1) n T and [0 1) n \D = f0g, fr if 0 6= d 2 [0 1) n \D we wuld have ; T \ (d + T ) > 0whichcntradicts measure{disjintness. Suppse that the inductin hypthesis is true fr all g 0 g. Tprve itfrg we argue by cntradictin. S suppse that the hypthesis is false fr g, s either ; T \ (g +[0 1) n ) > 0 but g +[0 1) n 6 T,rD \ (g +[0 1) n ) 6 Z n.

10 108 J. C. LAGARIAS AND Y. WANG If T g +[0 1) n but there exists a d 2D \ (g +[0 1) n ) with d 62 Z n,then again ; T \ (d + T ) > 0,acntradictin. S (3.9) ; T \ (g +[0 1) n ) > 0andg +[0 1) n 6 T: S Ntice that d2db l (T + d) cvers g +[0 1)n (in measure{disjint fashin). S there exists at least ne 0 6= d 2D B l such that (3.10) ; ; (T + d) \ (g +[0 1) n ) > 0 r equivalently T \ (g ; d +[0 1) n ) > 0: This gives rise t tw cases: d 2 Z n and d 62 Z n. If d 2 Z n,theng;d 2 Z n and (3.10) implies g;d 2 Z n +.Sg;dgand hence T g ; d +[0 1) n,which tgether with (3.9) cntradicts measure{disjintness. Nw suppse that d 62 Z n. Let d = d B k d k where all d i 2D and d k 6= 0. If k 1 then clearly d ; d i 2 Z n + and jd ; d i j 1 1 fr all d i. S each d i 2 g i +[0 1) n fr sme g i g and hence g i 2 Z n. But this implies d 2 Z n,a cntradictin. Therefre k = 0 and d 2D. Furthermre d 2 g +[0 1) n because if nt then there wuld be a g 0 g such thatd 2 g 0 +[0 1) n,which again wuld give d 2 Z n and hence a cntradictin. We nwshwthatthiscntradicts the measure{disjintness cnditin. Ntice that any 0 6= d 2D B l satisfying (3.10) must lie in g +[0 1) n, s there exists exactly ne such 0 6= d 2D B l. Hence T [ (T + d) g +[0 1) n : Let f be the unique vectr in d +[0 1) n \ Z n. f ; g 2D. Since [0 1] n T it fllws that ; (T + d) \ (T + f ; g) > 0 S f ; g is a zer{ne vectr, a cntradictin. S we have prved ur hypthesis (i) and (ii), and Claim 3 fllws. Finally, we prve that (E ;E) \ (DB 1 ;D B 1 )=f0g. Suppse nt, then there exist f 1 f 2 2Eand g 1 g 2 2DB 1 such that f 1 ; f 2 = g 1 ; g 2 6= 0. S f 1 + g 2 = f 2 + g 1 and hence ; (T + g 1 ) \ (T + g 2 ) > 0: This cntradicts the measure{disjintness prperty (2.8). 4. Diagnal Case: Zer{One Plynmials Let B =diag(b 1 ::: b n ) be an expanding nnnegative diagnal matrix. We nw shw that the criterin f Therem 3.2 fr (B D) t be nnnegative and minimal feasible can be refrmulated in terms f plynmial factrizatins f zer{ ne plynmials. Let X be a nite subset f Z n + := Zn \ R n + and assign t it the generating plynmial (4.1) X p X (z) =p X (z 1 ::: z n ):= z d d2x in which z d := z1 d1 zd2 2 zdn n.suchaplynmial is just a zer{ne plynmial,i.e. p X (z) 2 Z[z 1 ::: z n ] with all cecients zer r ne. Accrding t Therem 3.2 we have (4.2) T (B D )= [ f2e(f +[0 1] n )

11 NONNEGATIVE RADIX REPRESENTATIONS FOR THE ORTHANT R n fr sme nite EZ n + such that (E ;E) \ (DB 1 ;D B 1 )=f0g. Set (4.3) B = n[g 1 g 2 g n ] T : 0 g i <b i : Nw the functinal equatin (3.7) can be encded using (4.2) as the plynmial functin identity (4.4) p E (z B )p B (z) =p E (z)p D (z) where z B := (z1 b1 ::: zbn n ). Mre generally, ne can cnsider the plynmial factrizatin identity (4.5) f(z B )g(z) =f(z)h(z) where f(z), g(z), h(z) are all zer{ne plynmials. Classifying all slutins f (4.5) seems an interesting prblem. It includes many slutins ther than thse cming frm minimal feasible digit sets. Fr example, special cases f this identity arise frm integer self{ane tiles T (B D) (as dened in [10]) that are unins f lattice cubes these include examples fr which (B D) is nt feasible, with [0 1] n 6 T (B D). Here we will nly prve a result that classies all slutins t (4.5) that satisfy sme stringent side cnditins, which hwever cver all cases (4.4). Let D = diag(d 1 d 2 ::: d n ) be a diagnal matrix in which all d i are integers with d i > 1, and let f(z) = P g2x a gz g be a plynmial. We dene (4.6) where X [f] D (z) := a 0 gz g g2x a 0 ag if g =[g g = 1 ::: g n ] T with all g i <d i, 0 therwise: Therem 4.1. Let D = diag(d 1 ::: d n ) in which all d i > 1 are integers. Suppse f(z), g(z) h(z) are allzer{ne plynmials with f(0) =g(0) =h(0) =1satisfying (4.7) Suppse further that f(z D )g(z) =f(z)h(z): (i) [g] D (z) =g(z) and g(z) has n factrs f multiplicity greater than 1. (ii) Fr any f 2 Z n +, any zer{ne vectr d 6= 0, and any integer m 0, the cecient f either z Dm f r z Dm (f+d) is 0 in f(z). Then there exist zer{ne plynmials g 0 (z) g 1 (z) ::: g m (z) such that (4.8) (4.9) (4.10) g(z) = g 0 (z)g 1 (z) g m (z) h(z) = g 0 (z)g 1 (z D ) g m (z Dm ) f(z) = my iy g i (z Dj ): i=1 j=0 Remark. (1). It is pssible that sme f the g i (z) = 1. (2). The rather strange{ lking hypthesis (ii) actually encdes a nn{verlapping prperty that the set E in (4.3) pssesses. We will derive this therem recursively frm the fllwing lemma.

12 110 J. C. LAGARIAS AND Y. WANG Lemma 4.2. Assume that f(z) g(z) h(z) are zer{ne plynmials satisfying (4.11) f(z D )g(z) =f(z)h(z) and all the ther hypthesis f Therem 4.1. Then (4.12) and (4.13) (4.14) g(z) =[f] D (z)[h] D (z) f(z) = [f] D (z)f 1 (z D ) h(z) = [h] D (z)h 1 (z D ) where f 1 (z) and h 1 (z) are zer{ne plynmials. Prf. We write f(z) = a 0 (z)z Df0 + a 1 (z)z Df1 + + a k (z)z Dfk h(z) = b 0 (z)z Dh0 + b 1 (z)z Dh1 + + b k (z)z Dhl where all a i (z) andb i (z) satisfy [a i ] D (z) =a i (z) 6= 0 [b j ] D (z) =b j (z) 6= 0 and f 0 = h 0 = 0 and the ff i g (resp. fh i g) are all distinct nnnegative vectrs. Nte als that (4.15) a 0 (z) =[f] D (z) b 0 (z) =[g] D (z): We rst shw that a 0 (z)b 0 (z) =g(z) which is (4.12). Since f(z) = 1 + fther termsg, (4.11) yields g(z) =[f(z D )g(z)] D =[f(z)h(z)] D =[a 0 (z)b 0 (z)] D : Thus if a 0 (z)b 0 (z) 6= g(z), then a 0 (z)b 0 (z) cntains sme mnmial z g+de having e 6= 0 and g =[g 1 g n ] T with g i <d i fr all i. Als since deg zi (a 0 (z)b 0 (z)) 2(d i ; 1) the vectr e must be a zer{ne vectr. Therefre, again frm (4.11), since g(z) =[g(z)] D by (i), the nly mnmial in f(z D ) that can prduce such a mnmial in f(z D )g(z) isz De. Thus z e must be a mnmial inf(z). But e is a nntrivial zer{ne vectr, and this cntradicts (ii), taking f = 0 d = e m=0. We next prve thatfrall1ik and 1 j l, (4.16) a i (z) =a 0 (z) b j (z) =b 0 (z): We arrange the vectrs ff i h j : 0 i k 0 j lg int a sequence ft 1 t 2 ::: t k+l+2 g in such away that if jt i j 1 < jt j j 1 then we necessarily have i<j. (Here jtj 1 dentes the sum f the crdinates f t, itsl 1 {nrm.) We prve (4.16) by establishing the fllwing hypthesis, by inductin n m: if t m = f i (resp. t m = h j ) then a i (z) =a 0 (z) (resp. b j (z) =b 0 (z)). The hypthesis is clearly true fr m = 1, since t 1 is either f 0 r h 0. Suppse that the hypthesis is true fr m 0 <m.nw, t m = f i fr sme 0 <i k r t m = h j fr sme 0 <j l. Ift m = f i,thenwe cnsider the term a i (z)b 0 (z)z Dfi in the expansin f f(z)h(z). We rst bserve thata i (z)b 0 (z) must be a zer{ ne plynmial because f(z)h(z) =f(z D )g(z) is. Next we bserve that because b 0 (0) = 1, all terms in a i (z)z Dfi are als terms in f(z)h(z) =f(z D )g(z). Hence z Dfi is a term in f(z D ) and s z fi is a term in f(z). We claim that g(z);a i (z)b 0 (z) must als be a zer{ne plynmial. If nt, then frm (4.11) there is a term z g+df in a i (z)b 0 (z), where f 6= 0 is a zer{ne vectr while g =[g 1 g n ] T satisfy

13 NONNEGATIVE RADIX REPRESENTATIONS FOR THE ORTHANT R n g i <d i fr all i. It fllws frm (4.11) that z D(f+fi) is a term in f(z D ), s z f+fi is aterminf(z). But z fi is als a term in f(z). This is a cntradictin. Therefre, bth a i (z)b 0 (z) andg(z) ; a i (z)b 0 (z) are zer{ne plynmials. Nw suppse that a i (z) 6= a 0 (z). Then g(z) ; a i (z)b 0 (z) 6= 0. S in rder fr (4.11) t hld there must be sme i 0 6= i j 0 6= 0 such that Df i 0 + Dh j 0 = Df i, f i 0 + h j 0 = f i. If i 0 > 0 and j 0 > 0, then bth f i 0 and h j 0 are ahead f f i in the sequence ft j g. S a i 0(z) = a 0 (z) andb j 0(z) = b 0 (z). But this wuld give a i 0(z)b j 0(z) =g(z) and hence a i (z)b 0 (z) = 0, which isacntradictin. Therefre i 0 = 0 (since j 0 6= 0), and s f 0 + h j 0 = h j 0 = f i. It fllws that g(z) =a i (z)b 0 (z)+a 0 (z)b j 0(z): But g(z) =a 0 (z)b 0 (z) andg(z) has n factrs with multiplicity greater than 1, s a 0 (z) andb 0 (z) must be relatively prime. On the ther hand, a i (z)b 0 (z) =g(z) ; a 0 (z)b j 0(z) =a 0 (z) ; b 0 (z) ; b j 0(z) : Hence a 0 (z)ja i (z), s by the nnnegativity b 0 (z)a i (z) =g(z) and thus a i (z) =a 0 (z), cntradicting ur assumptin. In the case f t m = h j, b j (z) =b 0 (z) isprved in the identical fashin. Thus (4.16) is prved. Finally (4.15) and (4.16) cmbine t prve (4.13) and (4.14). Prf f Therem 4.1. Let g 0 (z) =[h] D (z) and g (z) =[f] D (z) = g(z)=g 0 (z). Then frm Lemma 4.2 we have g (z D )f 1 (z D2 )g 0 (z)g (z) =g 0 (z)h 1 (z D )g (z)f 1 (z D ): This gives g (z D )f 1 (z D2 )=h 1 (z D )f 1 (z D ) and hence (4.17) f 1 (z D )g (z) =h 1 (z)f 1 (z): It is clear that f 1 (z) g (z) h 1 (z) als satisfy the assumptins f Lemma 4.2. Hence we may repeat the abve prcess by setting g 1 (z) =[h 1 ] D (z) etc. This prcess eventually gives us (4.8){(4.10). We nwgback t ur digit sets E B and D,which satisfy (4.4). Nw all the hyptheses f Therem 4.1 are satised fr f(z) g(z) h(z) equal t p E (z) p B (z), p D (z), respectively. Fr (i) clearly hlds fr p E (z) and prperty (ii) hlds as a cnsequence f Therem 3.2: D cntains all zer{ne vectrs and (E;E)\(D B 1 ; D B 1 )=f0g. Thus Therem 4.1 shws there exist nite subsets B 0 B 1 ::: B m f Z n + such that (4.18) (4.19) (4.20) p B (z) = p B0 (z)p B1 (z) p Bm (z) p D (z) = p B0 (z)p B1 (z B ) p Bm (z Bm ) p E (z) = my iy p Bi (z Bj ): i=1 j=0 Using these frmulae, we classify all nnnegative and minimal feasible pairs (B D) when B is a diagnal matrix. Therem 4.3. Suppse that B =diag(b 1 b 2 ::: b n ), with all b i > 1 and DR n +. Then (B D) is minimal feasible if and nly if b i 2 Z fr all 1 i n and D is the Cartesian prduct f ne{dimensinal digit sets D 1 D 2 ::: D n, i.e.

14 112 J. C. LAGARIAS AND Y. WANG (4.21) D = D 1 D 2 D n where (b i D i ) are minimal feasible fr all 1 i n. Prf. We rst prve the \nly if" part f the therem. Suppse that (B D) is nnnegative and minimal feasible. Then by Therem 3.2 there exists a unique diagnal matrix S = diag(s 1 s 2 ::: s n ), s i > 0, such that D = S(D) Z n + cntains all zer{ne vectrs, and by the discussin abve (4.18){(4.20) hld. Nw n B = [g 1 g 2 ::: g n ] T : 0 g i <b i fr all i s (4.22) p B (z) = ny q i (z i ) i=1 where q i (z) =1+z i + z 2 i + + zbi;1 i is a plynmial in z i alne. Nw set ' i j (z i ):=g: c: d:(q i (z i ) p Bj (z)) nting that ' i j (z i ) divides q i (z i )sisaplynmialinz i alne. It is determined up t a multiplicative cnstant factr, and since p B (0) = 1, (4.18) shws that all p Bj (0) = 1, hence ' i j (0) 6= 0 s we may nrmalize it by requiring ' i j (0) = 1. Nw (4.18) and (4.22) tgether imply that (4.23) p Bj (z) =c 0 n Y i=1 ' i j (z i ) fr sme cnstants c 0, and taking z = 0 shws c 0 =1. Nw (4.23) implies that each ' i j (z i ) is a zer{ne plynmial, because p Bj (z) is, and each fthe' i j (z i ) depends n z i nly. Wenw substitute the expressin (4.23) int (4.19) and btain (4.24) p D (z) = my ny j=1 i=1 ' i j (z bj i i )= n Y i=1 m Y j=1 ' i j (z bj i i ) : Q m Next we bserve that j=1 ' i j(z bj i i ) is a zer{ne plynmial f z i,fr' i j (z i )isa zer{ne plynmial f degree at mst b i ;1 it therefre is a generating plynmial p D i (z i ) fr sme subset D i Z. Nw(4.19)becmes (4.25) p D (z) = ny p D i (z i ) i=1 and hence D = D1 D 2 D n : Thus D = S ;1 (D )=D 1 D 2 D n where D i = s ;1 i D i, and each (b i D i ) is nnnegative. Ntice that T (B D) =T (b 1 D 1 ) T (b 1 D 2 ) T (b 1 D n ) and T (B D) has Lebesgue measure equal t the prduct f the ne{dimensinal Lebesgue measures f T (b i D i ). Thus the feasibility f(b D) implies that f each (b i D i ), whence jd i jb i.hwever jd 1 jjd n j = j det(b)j = b 1 b n and it fllws that jd i j = b i,seach(b i D i ) is minimal feasible. Wenwprve the \if" part f the therem. If (b i D i )isnnnegative and minimal feasible fr all 1 i n, then clearly every x 2 R n + has a radix expansin using

15 NONNEGATIVE RADIX REPRESENTATIONS FOR THE ORTHANT R n base B and digits frm D = D 1 D 2 D n. Hence (B D) must be feasible, and nnnegativity and minimality are bvius. 5. General Case The results f Sectin 2 shwed that if (A D) is nnnegative and feasible then A = BP where B is a diagnal matrix and P is a permutatin matrix. We nw classify all nnnegative minimal feasible pairs (A D) fr such A. Fr any permutatin f f1 2 ::: ng, let P be the permutatin matrix 1 (P ) i j = if j = (i), 0 therwise. Cnsider the cyclic decmpsitin f, = 1 2 k int disjint cycles, e.g (1 3 4)(2 5)(6) represents : Fr any i 2f1 2 ::: ng write i 2 j =(j 1 ::: j l )ifsmej m = i. Let A = BP where B = diag(b 1 :::b n ). Nw A splits up int blcks crrespnding t the cycle decmpsitin =( 1 )( 2 ) ( k )f. Set (5.1) where b (i) j (5.2) = b j if j 2 i and b (i) j B i =diag(b (i) 1 b(i) 2 ::: b(i) n ) = 1 therwise. Then A = BP =(B 1 P 1 )(B 2 P 2 ) (B n P n ) and the matrices B i P i all cmmute pairwise. Fr example, if >0 is arbitrary, then (5.3) A = = diag( 3 4 )P (1 3)P (2) = B (1 3) P (1 3) B (2) P (2) where B (1 3) = diag( 1 4 ) and B (2) = diag(1 3 1). Lemma 5.1. Let A = BP where is an n{cycle, i.e. a cycle f length n, and B =diag(b 1 ::: b n ) with b = b 1 b 2 b n 2 Z. Suppse that (A D) is nnnegative and minimal feasible. Then there exists an integer 1 k n such that all digits d 2D lie n the x k {axis, i.e. d = de k fr sme d 2 R + fr all d 2D. Prf. We rst bserve thata n = bi and since (bi D A n ) is als nnnegative and minimal feasible, by Therem 4.3 the set D A n = D + A(D)++ A n;1 (D) must be the Cartesian prduct f n ne{dimensinal minimal feasible digit sets. S there must be exactly n(b ; 1) nnzer digits in D A n that lie n ne f the crdinate axes. Since the actin f A is t permute crdinate axes, each f these digits has the frm A k d fr a digit d 2 D lying n sme crdinate axis and 0 k n ; 1. Nw 0 2Ds there are b ; 1 nnzer digits d 2D. Hence we cnclude that all digits d 2Dmust lie n the n crdinate axes. It remains t shw thatalld2dmust lie n the same crdinate axis, i.e. all d i = d i e l fr sme crdinate axis vectr e l and sme d i 2 R +. We argue by cntradictin. Suppse nt, then fr each 0 k<nthe set A k (D) includes vectrs

16 114 J. C. LAGARIAS AND Y. WANG n at least tw dierent crdinate axes. Nw chse a d 2 A k (D) with 0 k<n such that n S jd n;1 (5.4) j 1 =min jdj 1 : d 2 j=0 Aj (D) : Suppse that d = d e m and chse any d 0 = d 0 e l 2 A k (D) such thatl 6= m. Bth d d 0 2D A n and d is rthgnal t d 0, while Therem 4.3 shws that D A n is a Cartesian prduct f crdinate axes, hence d + d 0 2D A n. Thus there exists a representatin (5.5) X n;1 d e m + d 0 e l = A j d j all d j 2D: j=0 Since each A j d j lie n a crdinate axis, (5.4) implies that ne f the terms n the right{hand side f (5.5) must be d. It is necessarily A k d k = d, hence d 0 e l = X 0j<n j6=k A j d j all d j 2D: This cntradicts the fact that all b n expansins in D A n must be distinct. Thus all d 2Dmust lie n the same crdinate axis. At last we can btain the desired classicatin. Therem 5.2. Let A be an expanding nnnegative matrix and DR n +. Then (A D) is minimal feasible if and nly if the fllwing cnditins hld: (i) A = BP where B =diag(b 1 b 2 ::: b n ) and P is a permutatin matrix. If has the cyclic decmpsitin = 1 2 k,then (5.6) b j := Y i2 j b i 2 Z + 1 j k with each b j 2. (ii) There exist ne{dimensinal digit sets D 1 D 2 ::: D k R +,withjd j j = b j, such that (b j D j) are nnnegative and minimal feasible fr all 1 j k, and (5.7) D = D 1 e l1 + D 2 e l2 + + D k e lk fr crdinate vectrs e lj with l j 2 j fr all 1 j k. Remark. This therem immediately implies Therem 1.2, since fr every A satisfying (i), there clearly exists a set D satisfying (ii). Prf. We rst prve the \nly if" part. Suppse that (A D) is nnnegative and minimal feasible. By Lemma 2.3 A = BP = BP 1 P 2 P k : Fr each 1 j k, set R j := n[x 1 x 2 ::: x n ] T : x i =0ifi 62 j and R j + := R j \ R n +. Next set D j := D\R j. Cnsider the decmpsitin (5.2) f A: A = BP =(B 1 P 1 )(B 2 P 2 ) (B k P k ): It fllws frm the nnnegativity f(a D) anda(r j )=R j that each x 2 R j + has a radix expansin using nly digits in D j. Ntice that fr any d 2D j ne has Ad =(B j P j )d:

17 NONNEGATIVE RADIX REPRESENTATIONS FOR THE ORTHANT R n Thus (B j P j D j )whenviewed as acting n R j (5.8) c j := jd j jjdet(b j P j )j = b j: is a feasible pair, hence Let m 1 such that A m is a diagnal matrix, and recall the pair (A m D A m )is minimal feasible using Lemma 2.1. Clearly hence (5.9) D A m \ R j = D j B j P j m jd A m \ R j j = jd j B j P j m j = jdj A m j = cm j : But because A m is diagnal, it fllws frm Therem 4.3 that its digit set D A m is a direct sum f ne{dimensinal digit sets, hence in particular (5.10) D A m D 1 A m + + Dk A m = D1 B 1 P 1 m + + Dk B k P k m which is a direct sum because all summands lie in mutually rthgnal subspaces. This gives jd A m j = ; b 1 b 2 b k m c m 1 c m 2 cm k using (5.9). Cmparisn with (5.8) then implies that (5.11) b j = c j 2 Z 1 i k and clearly b j > 1. Mrever, it nw fllws frm (5.10) and (5.11) that (5.12) D A m = D 1 B 1 P 1 m + + Dk B k P k m : Nw each(b j P j D j )viewed as acting n R j is a nnnegative minimal feasible pair, using (5.11), and the hypthesis f Lemma 5.1 are satised fr it. Hence there exists an l j 2 j such thatd j = D j e lj where D j R and each (b j D j)isa nnnegative minimal feasible ne{dimensinal pair. T establish (ii) it remains t shw that (5.7) hlds, i.e. D = D ~ where ~D = D 1 + D D k : Tprve this it suces t shw that D ~ D, because they have the same cardinality b = b 1 b k.we cnsider the rthgnal prjectin peratr i : R n ;! R i and examine the set i (D), with elements cunted with multiplicity. If weshw that all elements f i (D) lieind i = D i e i fr 1 i k, then D ~ D.We will actually shw the strnger result: (5.13) i (D) isd i with all elements f multiplicity b=b i. Nw R i is an invariant subspace f A, and i cmmutes with B i and P i. In particular, using (5.12) and ignring multiplicity, (5.14) ; i (D) A m = i(d A m ) ; = i D 1 A m + D2 A m + + Dk A m = D i A m = X nm;1 j=0 A j d (i) j : all d (i) j 2D i : If we cunt multiplicity, then (5.12) implies that ; i (D) A m := i(d)+a i (D)++ A m;1 i (D) (5.15)

18 116 J. C. LAGARIAS AND Y. WANG is just D i A m with all elements cunted with multiplicity (b=b i ) m. Nw the rst part f the prf f Lemma 5.1 applies t the set (5.15) t cnclude that every element f i (D) lies n a crdinate axis. T prceed, we determine the elements f D i A m that lie n sme crdinate axis. Nw i is a cycle f rder q,say i =(r 1 r 2 ::: r q )withr 1 = l i,say, and necessarily q divides m. The decmpsitin (5.14) factrizes as X X q;1 A jm=q j=0 l=0 A ql d (i) ql+j where the inner sum invlves nly crdinate vectrs alng the axis e r1 = e li. In particular ( i (D)) A m cntains vectrs n crdinate axis e rj with cecients lying in the sets (5.16) S j := na j ; Di + D i b i + + D i (b i ) m k ;1 where a 1 =1anda j = b r1 b r2 b rj;1 fr 1 <j<q; 1, and b i = b r 1 b rq 2is an integer. Suppse nw that i (D) cntains sme elements d 6= de r1 fr any d 2 R. It then lies n sme crdinate axis e rj, and necessarily has a cecient frm the set (5.16). In particular any f its pwers A j d that lie n the axis e r1 have cecients in the set D i b i + + D i(b i ) m k ;1 : Hence they can never appear in the expansin f any elements in ( i (D)) A m f the frm d 0 e r0 with d 0 2D i.thus all representatins f such vectrs in the expressin (5.14) must be f the frm (5.17) d 0 e r0 = d 0 e r0 + A(0)++ A k;1 (0): Hwever 0 must ccur with its crrect multiplicity b=b i in i(d), in rder t get exactly (b=b i )m cpies f 0 in ( i (D)) A m, since all terms in (5.14) are nnnegative. Thus the number f representatins f d 0 e r0 n the right side f (5.17) is (b=b i ) m;1 (multiplicity fd 0 e r0 in i (D)): This number is (b=b i )m, hence d 0 e r0 has multiplicity b=b i in i(d) fr all d 0 2D i. This cunt exhausts i (D), and prves (5.13). Hence D = ~ D,prving (ii). We nw nish the prf by prving the \if" part f the therem. Accrding t Lemma 5.1, if we viewb j P j and D j = D j e lj as lying n R j, then (B j P j D j ) are nnnegative and minimal feasible, and T (A D) =T (B 1 P 1 D 1 ) T (B k P k D k ) if the crdinates are suitably numbered. Hence (A D) is feasible, and nnnegativity and minimality are clear. Example 5.1. The matrix A := := BP := given in (5.3) satises cnditin (i) f Therem 5.2 fr all >0. By Therem 5.2 (A D) is feasible fr D = f ge 1 + f0 1 2ge 2 3 5

19 NONNEGATIVE RADIX REPRESENTATIONS FOR THE ORTHANT R n because f g is feasible fr base 4 =4andf0 1 2g is feasible fr base 3. We can see this mre easily frm A 2 = diag(4 9 4) and D A 2 = f ge 1 + f0 1 ::: 8ge f ge 3 which is a direct sum f three ne{dimensinal feasible digit sets. B = diag( 3 4 ) is an expanding matrix nly fr 1 <<4. References Nte that 1. C. Bandt, Self-similar sets 5. integer matrices and fractal tilings f R n, Prc. Amer. Math Sc. 112 (1991), 549{562. MR 92d: M. Barnsley, Fractals everywhere, Academic Press, MR 90e: L. Carlitz and L. Mser, On sme special factrizatins f (1 ; X n )=(1 ; X), Canad. Math. Bull. 9 (1966), 421{426. MR 34: W. Gilbert, Gemetry f radix expansins, in: The Gemetry Vein: the Cxeter Festschrift (1981), 129{139. MR 83j: K. Grchenig and A. Haas, Self{similar lattice tilings, J. Furier Analysis 1 (1994), 131{ R. Kenyn, Self-similar tilings, Ph.D thesis, Princetn University (1990). 7. R. Kenyn, Self-replicating tilings, in: Symblic Dynamics and Applicatins (P. Walters, ed.) Cntemprary Math. vl. 135 (1992), 239{264. MR 94a: D. E. Knuth, The art f cmputer prgramming: vlume 2. Seminumerical algrithms, Addisn{Wesley, (See Chapter 4.1, exercise 20{24.) MR 44: J.C. Lagarias and Y. Wang, Self-ane tiles in R n, Advances in Math., t appear. 10. J.C. Lagarias and Y. Wang, Integral self-ane tiles in R n I. Standard and nnstandard digit sets, J. Lndn Math. Sc., t appear. 11. D. W. Matula, Basic digit sets fr radix representatins, J. Assc. Cmput. Mech. 4 (1982), 1131{1143. MR83k: A. M. Odlyzk, Nnnegative digit sets in psitinal number systems, Prc. Lndn Math. Sc. 37 (1978), 213{229. MR 80m: A. Vince, Replicating Tesselatins, SIAM J. Discrete Math. 6, n. 3 (1993), 501{521. MR 94e:52023 AT&T Bell Labratries, 600 Muntain Avenue, Murray Hill, New Jersey address: jcl@research.att.cm Schl f Mathematics, Gergia Institute f Technlgy, Atlanta, Gergia address: wang@math.gatech.edu