ON THE ENTRY SUM OF CYCLOTOMIC ARRAYS. Don Coppersmith IDACCR. John Steinberger UC Davis

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006), #A26 ON THE ENTRY SUM OF CYCLOTOMIC ARRAYS Don Coppersmith IDACCR John Steinberger UC Dvis Received: 3/29/05, Revised: 0/8/06, Accepted: 0/0/06, Published: 0/3/06 Abstrct A cyclotomic rry is n rry of numbers tht cn be written s liner combintion of lines of s running prllel to the coordinte xes nd running the full length of the rry. In this pper we show tht the sum of the entries of cyclotomic rry with nonnegtive integer entries is nonnegtive integer liner combintion of the sidelengths of the rry. Introduction A cyclotomic rry is n rry of numbers tht cn be written s liner combintion of lines of s running prllel to the coordinte xes nd running the full length of the rry. Figure shows for exmple 2-dimensionl cyclotomic rry of size 3 4 nd how it is obtined s liner combintion of lines of s. Figure 2 shows similr exmple of 3-dimensionl cyclotomic rry. The purpose of this pper is to prove the following theorem: Theorem. The sum of the entries of cyclotomic rry with nonnegtive integer entries is nonnegtive integer liner combintion of the sidelengths of the rry. Lm nd Leung [6 prove the specil cse of Theorem when the two smllest sidelengths of the rry re coprime (Corollry 2 in our pper). Their pper, however, is written in the slightly different context of vnishing sums of roots of unity. The connection between vnishing sums of roots of unity nd cyclotomic rrys is briefly discussed below. A more complete discussion cn be found in [9 (where the term cyclotomic rry is lso coined). We shll refer to the lines of s in n rry s fibers. Thus n rry is cyclotomic if nd only if it cn be written s liner combintion of fibers. It is shown in [9 tht ny integervlued cyclotomic rry cn lwys be written s n integer liner combintion of fibers,

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006), #A26 2 - Figure : A 3 4 cyclotomic rry (on the left) shown decomposed s liner combintion of lines of s. Blnk entries denote zeroes. Figure 2: A 2 2 2 cyclotomic rry shown decomposed s liner combintion of lines of s; shded cubes denote s, other entries re 0. from which it follows tht the entry sum of n integer-vlued cyclotomic rry is lwys n integer liner combintion of the sidelengths of the rry. Theorem shows this integer liner combintion cn be written with nonnegtive coefficients if the rry lso hppens to be nonnegtive. Theorem is trivil for -dimensionl cyclotomic rrys, which hve ll their entries equl. It is lso esy for 2-dimensionl cyclotomic rrys, s it turns out tht ny nonnegtive 2- dimensionl cyclotomic rry is just positive sum of fibers [9. The problem only becomes interesting for 3-dimensionl rrys nd higher, s nonnegtive 3-dimensionl cyclotomic rrys or higher re not lwys positive sums of fibers (such s the rry of Fig. 2). If one wishes one cn restrict one s ttention to rrys tht re miniml, in the sense of not being decomposble s the sum of two other (nonzero) cyclotomic rrys with nonnegtive integer entries, since if Theorem is true for miniml rrys then it obviously holds for ll rrys. However miniml cyclotomic rrys of 3 or more dimensions cn hve surprisingly complicted structures. For exmple they my contin entries tht re rbitrrily lrge, even superpolynomilly lrge compred to the volume of the rry [9. It is therefore not of much use to restrict one s ttention to miniml cyclotomic rrys. An exception is Lm nd Leung s proof of Theorem for rrys whose two smllest sidelengths re coprime, which cn be obtined simply by giving lower bound on the entry sum of non-fiber miniml cyclotomic rrys (see Corollries nd 2 in our pper). Another promising but ultimtely doomed pproch for proving Theorem, s noted by Lm nd Leung, is to show tht ny integervlued nonnegtive cyclotomic rry hs representtion s n integer liner combintion of fibers where the sum of the coefficients of fibers prllel to the j-th coordinte direction is nonnegtive for ech j. In fct, even the 2 2 2 rry of Fig. 2 does not dmit such representtion, s the reder my verify.

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006), #A26 3 As just mentioned, Theorem for 2-dimensionl rrys is consequence of the fct tht nonnegtive 2-dimensionl cyclotomic rrys re positive sums of fibers (or lterntively, tht the only miniml 2-dimensionl cyclotomic rrys re fibers). A proof of this cn be found in [9 or cn be deduced from Proposition below. To wrm up we give slightly different rgument showing Theorem for 2-dimensionl rrys; this rgument is closer in structure to our proof for n-dimensionl rrys. If A is cyclotomic rry we write [A for the sum of the entries, [A + for the sum of the positive entries nd [A for the sum of the negtive entries in bsolute vlue (so [A = [A + [A ). A j-fiber mens fiber prllel to the j-th coordinte direction nd j-lyer mens n rry lyer perpendiculr to the j-th coordinte direction (thus j-lyer of n rry of size n is n rry of size j j+ n ). We index coordintes strting t 0 insted of t, so the j-th coordinte of cyclotomic rry of size n is number in Z j = {0,, 2,..., j }. We first note defining feture of cyclotomic rrys: Proposition. The difference of two j-lyers of cyclotomic rry of dimension n > is cyclotomic rry of dimension n. Proof. Let A be cyclotomic rry of size n. Fix some representtion of A s liner combintion of fibers. When we tke the difference of two j-lyers of A the contribution of j-fibers to those lyers cncels, leving only the contributions of i-fibers for i j. Thus the result of the difference is cyclotomic rry of size j j+ n. Thus two -lyers or two 2-lyers of 2-dimensionl cyclotomic rry differ only by n dditive constnt, since -dimensionl cyclotomic rrys hve ll their entries equl. From this we cn esily find proof of Theorem for 2-dimensionl cyclotomic rrys. The nottion Z + (, 2 ) mens the set of nonnegtive integer liner combintions of, 2 : Proposition 2. If A is nonnegtive integer-vlued cyclotomic rry of size 2 then [A Z + (, 2 ). Proof. Let A 0,..., A 2 be the 2 2-lyers of A, nd let A be 2-lyer of A such tht [A = min([a r : r Z 2 ). Also let A r = A r A for ll r Z 2. We hve [A = [A 2 + r Z 2 [A r. () By Proposition ech A r is -dimensionl integer-vlued cyclotomic rry of size, so [A r is multiple of for every r (recll tht the entries of -dimensionl cyclotomic rry re ll equl). But [A r = [A r [A 0 so [A r is nonnegtive multiple of for ll r. It thus directly follows from () tht [A Z + (, 2 ).

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006), #A26 4 3 2 b b 2 b 3 Figure 3: Inflting n rry. t 3 t 2 t s 3 s 2 r r 3 s 2 r Figure 4: Another infltion. (Note tht the sme type of proof cnnot work for 3-dimensionl cyclotomic rrys becuse 2-dimensionl integer-vlued cyclotomic rry A cn hve [A 0 but [A / Z + (, 2 ), such s the rry of Fig..) Some interesting pplictions of Theorem cn be found by exhibiting cyclotomic rrys whose entry sums re not obviously nonnegtive integer liner combintion of the sidelengths. A simple wy to construct cyclotomic rrys is to use process clled infltion. Let A i,...,i n denote the (i,..., i n )-th entry of n rry A of dimension n. Formlly, cyclotomic rry A of size n is sid to be n inflte of cyclotomic rry A of size n if there re functions κ : Z Z,..., κ n : Z n Z n such tht A i,...,i n = A κ (i ),..., κ n(i n) for ll (i,..., i n ) Z Z n (this notion of infltion differs slightly from the one defined in [9, where the functions κ j re required to be surjections). The bsic ide behind infltion is shown in Fig. 3. It is esy to check tht the infltes of cyclotomic rrys re gin cyclotomic. The sum of the entries of n inflte of the Fig. 2 cyclotomic rry is equl to b b 2 b 3 + ( b )( 2 b 2 )( 3 b 3 ) for some integers, 2, 3, b, b 2, b 3 with 0 b, 0 b 2 2, 0 b 3 3. It follows from Theorem tht b b 2 b 3 + ( b )( 2 b 2 )( 3 b 3 ) Z + (, 2, 3 ) (2) for ll integers, 2, 3, b, b 2, b 3 such tht 0 b, 0 b 2 2, 0 b 3 3. The inclusion (2) is not esy to verify without Theorem nd this pper is, s fr s we know, the first plce it hs been noted. Considering more generlly the infltes of odd-dimensionl

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006), #A26 5 Figure 5: The two fiber tilings of 5 4 rry. The shded strips represent the fibers used. 2 2 rrys with two entries of t opposing corners of the rry nd entries of 0 elsewhere (which re esily verified to be cyclotomic), we get tht n b i + i= n ( i b i ) Z + (,..., n ) (3) i= for ll integers,..., n nd b,..., b n such tht 0 b i i for ll i, for ll odd n. At the end of this section we give self-contined proof of (3) tht does not rely on cyclotomic rrys. Any number of identities of the type (2) nd (3) cn be found using Theorem. For nother exmple, it follows from considering infltes of the 3 3 3 cyclotomic rry on the left of Fig. 4 tht t (r s + r s 2 + r 2 s ) + t 2 (r 2 s + r 2 s 3 + r 3 s 3 ) + t 3 (r s 2 + r 3 s 2 + r 3 s 3 ) Z + (r, s, t) (4) for ll integers r, r 2, r 3, s, s 2, s 3, t, t 2, t 3 0, where r = r + r 2 + r 3, s = s + s 2 + s 3, t = t + t 2 + t 3. We hve lso found n musing ppliction of Theorem which concerns the notion of fiber tiling of n rry. A fiber tiling is collection F of fibers in n rry of size n such tht the sum of ll the fibers in F is the ll s rry (put nother wy, F is fiber tiling if nd only if the supports of the fibers in F prtition the set Z Z n ). There re only two kinds of two-dimensionl fiber-tilings: the tiling by horizontl fibers nd the tiling by verticl fibers (Fig. 5). Three-dimensionl fiber tilings re only slightly more vried, since these cn lwys be decomposed s sndwich of two-dimensionl fiber tilings, while four-dimensionl fiber tilings strt showing better diversity (in prticulr, four is the first dimension for which fibers in ll the coordinte directions cn pper simultneously in the sme tiling, if we exclude one-dimensionl tilings). Theorem gives us: Theorem 2. Let F be fiber tiling of n rry of size n where n 2. Then the number of j-fibers in F is in Z + (,..., j, j+,..., n ). Proof. Let F 0 denote ll i-fibers in F, i j, tht re contined in the first j-lyer of the rry, nd let A be the 0- j j+ n cyclotomic rry obtined by

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006), #A26 6 dding ll the fibers in F 0. Note tht the j-fibers of F re in -to- correspondence with the 0 entries of A. Let B denote the ll s j j+ n rry. Becuse the dimension of B is greter thn or equl to (s we supposed n 2) B is lso cyclotomic rry, so B A is nother 0- j j+ n cyclotomic rry. But the number of j-fibers in F is the entry sum of B A, so is in Z + (,..., j, j+,..., n ) by Theorem. We cn lso mention s n ppliction of Theorem the originl result of Lm nd Leung [6 on vnishing sums of roots of unity: Theorem 3. (Lm nd Leung [6) Let ζ n be primitive n-th root of unity nd let n c i ζn i = 0 i=0 be vnishing sum of n-th roots of unity where the c i s re nonnegtive integers. n i=0 c i is nonnegtive integer liner combintion of the primes dividing n. Then We only sketch the reduction from Theorem 3 to Theorem, since detiled ccount of the reltionship between cyclotomic rrys nd vnishing sums of roots of unity cn be found in [9. Firstly Theorem 3 cn be reduced to the cse where n is squrefree (becuse Φ np (x) = Φ n (x p ) if p is prime dividing n, where Φ m (x) is the m-th cyclotomic polynomil). Then if n = p p k is squrefree there is bijection between vnishing sums of n-th roots of unity nd cyclotomic rrys of size p p k, given by putting the coefficient of ζn i in the vnishing sum s the vlue of the entry with coordintes (i mod p,..., i mod p k ) in the rry (note tht under this bijection, j-fiber mps to regulr p j -gon in the complex plne, so the bijection essentilly sttes tht ny vnishing sum of roots of unity cn be obtined by ddition nd subtrction of regulr p-gons from one nother). Thus if the coefficients c i re nonnegtive integers it directly follows from Theorem tht n i=0 c i is in Z + (p,..., p k ). Conversely, Theorem 3 implies Theorem for rrys whose sides re distinct primes, though the proof technique of Lm nd Leung cn be more generlly dpted to prove Theorem for ll rrys whose two smllest sides re coprime (cf. Corollry 2). Lm nd Leung show some further pplictions of Theorem 3 to representtion theory nd there re mny other independent pplictions of vnishing sums of roots of unity (e.g. [3, 8, 0). Since the j-fibers of n n rry re the incidence vectors of cosets of the cnonicl copy of Z j in Z Z n, Theorem suggests the question of whether, more generlly, ny nonnegtive vector obtined s the integer combintion of cosets in finite belin group hs entry sum equl to some nonnegtive integer combintion of the sizes of the cosets used. This is flse; Fig. 6 shows n exmple where single element of G = (Z 3 ) 2 (Z 2 ) 3 is written s the difference of seven cosets of size 2 from five cosets of size 3, wheres / Z + (2, 3) ( similr exmple fits in the group G = (Z 3 ) 2 (Z 2 ) 2, but is hrder to drw due to overlp between the cosets of size 3). However Theorem does hve generliztion in the non-belin setting, essentilly due to Hertweck [5:

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006), #A26 7 f g c d e f g b c b e d Figure 6: An integer combintion of cosets of sizes 2 nd 3 in G = (Z 3 ) 2 (Z 2 ) 3 resulting in single element. Ech group of 3 dots connected by lines is coset of size 3 dded to the liner combintion nd ech pir of dots with the sme letter is coset of size 2 subtrcted from the liner combintion. The shded dot is the leftover element. Theorem 4. Let G be finite group nd let N,..., N k be norml subgroups of G such tht N N k = N N k. Then ny nonnegtive integer-vlued vector in Z G obtined s liner combintion of cosets of N,..., N k hs entry sum in Z + ( N,..., N k ). Proof. (Reducing from Theorem.) It obviously suffices to consider the cse where G = N N k. Then every element of G cn be uniquely written s product n n 2... n k where n i N i. For ech N i choose n rbitrry one-to-one mp f i from N i to Z Ni, nd define f : G Z N Z Nk by f(n n k ) = (f (n ),..., f k (n k )). If N i is coset of N i we cn write N i = n... n i N i n i+... n k for some n,..., n i, n i+,..., n k with n j N j for ll j (using the normlity of N,..., N k ) so f(n i) = f (n ) f i (n i ) Z Ni f i+ (n i+ ) f k (n k ) is coset of Z Ni in Z N Z Nk. Therefore every liner combintion of cosets of N,..., N k in G corresponds to liner combintion of fibers in n rry of size N N k nd Theorem 4 follows directly from Theorem. (Hertweck [5 did the equivlent generliztion for Lm nd Leung s result rther thn for Theorem, of which he ws unwre; he lso dpted Lm nd Leung s proof to the nonbelin setting insted of reducing the non-belin cse to the belin one.) The reder will hve noted tht the counterexmple of Fig. 6 uses digonl cosets of size 3 (for lck of better term). It seems hrd to construct counterexmple without using such digonl cosets. In this connection we offer up the following conjecture generlizing Theorem : Conjecture. Let G = Z Z n nd let v (Z + ) G be obtined s n integer liner combintion of incidence vectors of cosets of subgroups H,..., H m of G where ech H i is of the form J i J in for some subgroups J i,..., J in of Z,..., Z n respectively. Then the sum of the entries of v is in Z + ( H,..., H m ).

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006), #A26 8 Conjecture reduces to Theorem if we let m = n nd let H i = 0 Z i 0 for i n. If on the other hnd we set n = then Conjecture simply stipultes tht ny nonnegtive vector obtined s the integer liner combintion of cosets in cyclic group hs entry sum equl to some nonnegtive integer liner combintion of the sizes of the cosets used. It is noteworthy tht Conjecture cn be reduced to the cse where every J ik is either 0 or Z k : Proposition 3. Conjecture reduces to the cse where for ll i m ech J ik 0 or Z k. is either Proof. We cn ssume without loss of generlity tht ech i is prime power, since otherwise Z i cn be expnded s the direct product of cyclic groups of prime power order. It is now sufficient to show tht if = p α is prime power then there is -to- mpping f from Z to (Z p ) α such tht for ny subgroup J of Z nd ny j Z the imge f(j + j) of the coset J + j is direct product coset of (Z p ) α, nmely coset of the form J + j where J = J J α where ech J k is either 0 or Z p nd where j (Z p ) α. A simple mp f with this property is the rdix p mp, defined by setting the i-th coordinte of f(j) equl to the i-th digit of j written bse p. It is then esy to verify tht cosets of Z re mpped to direct product cosets of (Z p ) α under f, which completes the reduction. We finish the introduction with purely number-theoretic proof of (3). This proof is independent of the min result, so cn be sfely skipped if wished. We should still quickly mention, however, tht our min result is ctully stronger thn Theorem, s we hve shown tht Theorem lso holds for cyclotomic rrys with limited number of negtive entries. Theorem 6 in the next section gives the full sttement. Our proof of (3), s well s much else tht we do, relies on the following proposition: Proposition 4. (Buer [, Buer nd Shockley [2) Let 2 n be nturl numbers. Let λ n = gcd(,..., n ) nd let z > n /λ n n be divisible by λ n. Then z Z + (,..., n ). Proof. We do the proof by induction on n, s the conclusion obviously holds for n =. Let n 2 nd let λ n = gcd(,..., n ). Let u 0 be the lest integer such tht z u n 0 mod λ n. Notice u λ n /λ n. We hve z u n > n /λ n n u n n /λ n n (λ n /λ n ) n = n ( λ n )/λ n n ( λ n )/λ n = n /λ n n so tht, by induction, z u n Z + (,..., n ), which implies z Z + (,..., n ).

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006), #A26 9 Theorem 5. (Proof of (3)) Let n be odd nd let,..., n, b,..., b n be integers such tht 0 b i i for i n. Then n b i + i= n ( i b i ) Z + (,..., n ) (5) i= Proof. We do the proof by induction on n with the cse n = being obvious nd the cse n = 3 serving s our induction bsis. Assume therefore tht n = 3. We cn ssume without loss of generlity tht 2 3 nd tht 0 < b i < i since if b i = 0 or b i = i for some i the conclusion is obvious. We cn lso ssume tht b b 2 ( b )( 2 b 2 ) since otherwise we cn effect the chnge of vribles b = b, b 2 = 2 b 2. Put N = b b 2 b 3 + ( b )( 2 b 2 )( 3 b 3 ). Note tht N = b b 2 3 + (( b )( 2 b 2 ) b b 2 )( 3 b 3 ) b b 2 3 + (( b )( 2 b 2 ) b b 2 ). (6) The minimum of the function f(b, B 2 ) = ( B )( 2 B 2 ) B B 2 = 2 B 2 B 2 subject to the constrints B, B 2, B B 2 = b b 2 is ttined t B = b b 2, B 2 = (since 2 ) so f(b, B 2 ) ( b b 2 )( 2 ) b b 2 = ( b b 2 ) 2 for ll B, B 2 such tht B B 2 = b b 2. Since b, b 2 we get in prticulr tht so (6) implies tht ( b )( 2 b 2 ) b b 2 ( b b 2 ) 2 (7) N b b 2 3 + ( b b 2 ) 2. (8) Now put λ 2 = gcd(, 2 ), λ 3 = gcd(, 2, 3 ) (note N 0 mod λ 3, since the b b 2 b 3 cncels out with its negtive in the expnsion of ( b )( 2 b 2 )( 3 b 3 )). Let u 0 be the lest integer such tht N u 3 0 mod λ 2. Then u λ 2 /λ 3 nd lso u b b 2 since N b b 2 3 = (( b )( 2 b 2 ) b b 2 )( 3 b 3 ) 0 mod λ 2. From (8) we hve N u 3 (b b 2 u) 3 + ( b b 2 ) 2 (b b 2 u) 2 + ( b b 2 ) 2 = 2 /λ 2 + ( 2 2 /λ 2 ) u 2 2 /λ 2 + λ 2 ( 2 2 /λ 2 ) (λ 2 /λ 3 ) 2 = 2 /λ 2 + 2 (λ 2 λ 2 /λ 3 ) 2 /λ 2 > 2 /λ 2 2 which mens tht N u 3 Z + (, 2 ) by Proposition 4 nd thus tht N Z + (, 2, 3 ). This disptches the cse n = 3.

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006), #A26 0 Assume now tht n 5. Assume first tht l = m for some l m. Put = l = m. By symmetry we cn ssume tht ( b l )( b m ) b l b m (or else effect the chnge of vribles b i = i b i ). This implies tht b l b m 0. But then n b i + i= n n ( i b i ) = b l b m b i + ( b l )( b m ) i= = b l b m ( i= i l,m n i= i l,m b i + n ( i b i ) i= i l,m n ( i b i )) + ( b l b m ) i= i l,m n ( i b i ). Here the term b l b m ( n i= b i + n i= ( i b i )) is in Z + ({ i : i l, m}) by induction on n i l,m i l,m ( i b i ) is nonnegtive multiple of = l = m. nd the second term ( b l b m ) n i= i l,m Therefore n i= b i + n ll i, j, nd by symmetry tht < < n. Put i= i l,m i= ( i b i ) is in Z + (,..., n ). We cn thus ssume tht i j for β = gcd(,..., n ), X = n ( i b i ), Y = i= n b i, i= N = X( n b n ) + Y b n. Note tht X Y 0 mod β s n is odd. Our job is to prove N Z + (,..., n ). We cn ssume without loss of generlity tht X Y. If Y /β + β then we cn write Y = mβ + r for some m, r 0 such tht mβ > /β, nd we will hve N = Y n + (X Y )( n b n ) = r n + {mβ n + [(X Y )( n b n )} where the quntity in brckets {...} is multiple of β strictly greter thn n /β, so is in Z + (,..., n ) by Proposition 4. But then N Z + (,..., n ). So we cn ssume Y < /β + β. Recll X Y 0 mod β. If we ssume tht X Y > n /β n then X Y Z + (,..., n ) nd N = (X Y )( n b n ) + Y n Z + (,..., n ). We cn therefore ssume X Y n /β n. We now hve X = (X Y ) + Y ( n /β n ) + ( /β + β ) = (/β)( β)( n β + ) ( ) n

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006), #A26 b 2 2 b Figure 7: An 2 cyclotomic rry with [A = ( b )( 2 b 2 ) b b 2 nd [A = b b 2. so tht XY ( ) n. On the other hnd XY = n i= b i( i b i ) n i= ( i ), so we get n n ( i ). (9) i=2 But since < < n, n 5, we hve tht ( n 2 )( n ) > n, 2 < n 2, nd so n < ( 2 )( n 2 )( n ) n i=2 ( i ), contrdiction. Results We will strt by showing tht cyclotomic rry whose negtive entries hve smll but nonzero sum in bsolute vlue hs lrge overll entry sum. More precisely, we will show tht if A is n integer-vlued cyclotomic rry with two smllest sidelengths nd 2, 2, nd if [A > 0, then [A ( [A ) 2. (0) Note tht if [A is very smll compred to then (0) gives tht [A 2, which fits somewht well with Proposition 4 (see Corollries nd 2 nd the remrks therefter for the continution of this ide). Also note tht (7) is specil cse of (0), s illustrted by Fig. 7. Inequlity (0) ws first proved under different form by Lm nd Leung (cf. [6 Thm. 4.). Lm nd Leung in fct prove the stronger inequlity {A} + {A} ( {A} ) 2 () where {A} + is the number of positive entries of A nd {A} is the number of negtive entries of A, where () holds provided A is integer-vlued nd {A} > 0. We will not require this stronger version of (0). The proof tht we give here of Inequlity (0) is different from Lm

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006), #A26 2 (x, x 2, x 3 ) (y, y 2, y 3 ) Figure 8: An 5 5 6 rry with entries of 0 nd ±; + symbols denote entries of nd symbols denote entries of. Since this rry is orthogonl in R 5 5 6 to ll fibers it is lso orthogonl to ll cyclotomic rrys. Therefore if A is cyclotomic rry of size 5 5 6 we hve A x,x 2,x 3 A x,x 2,y 3 A x,y 2,x 3 A y,x 2,x 3 + A x,y 2,y 3 + A y,x 2,y 3 + A y,y 2,x 3 A y,y 2,y 3 = 0, i.e. (j,j 2,j 3 ) {0,} ( ) j +j 2 +j 3 A 3 j x +( j )y, j 2 x 2 +( j 2 )y 2, j 3 x 3 +( j 3 )y 3 = 0. nd Leung s proof, with the min dvntge tht it is shorter nd does not require induction on the dimension of the rry. Since the n-dimensionl cse of our proof of (0) is bit opque we will first give proof of the 2-dimensionl cse to illustrte the bsic ide (we will generlly be in the hbit of proving things severl times over in successive degrees of generlity, which we hope the reder will find more instructive thn nnoying). We strt by noting tht if A is n n cyclotomic rry then ( ) (j,...,jn) {0,} n j +...+j n A j x +( j )y,...,j nx n+( j n)y n = 0 (2) for ll pirs of coordintes (x,..., x n ), (y,..., y n ) in Z Z n. To understnd why (2) holds it suffices to look t Fig. 8 illustrting the 3-dimensionl cse. Proposition 5. Let A be n 2 cyclotomic rry such tht 2 nd such tht [A > 0. Then [A ( [A ) 2. Proof. Assume tht counterexmple exists with [A s smll s possible. If [A 2 then we cn dd either -fiber or 2-fiber to A (depending on whether the negtive entries of A re contined in common -fiber or not) such s to decrese [A by t lest nd increse [A by t most 2 while keeping [A > 0, thus obtining counterexmple with smller [A. It is therefore sufficient to consider the cse when [A =. In prticulr, we cn ssume tht A 0,0 = nd tht A i,j 0 for (i, j) (0, 0). Eq. (2) gives us tht A x,x 2 A y,x 2 A x,y 2 + A y,y 2 = 0 for ll (x, x 2 ), (y, y 2 ) Z Z 2. Summing up the reltions A 0,0 A i,0 A 0,j + A i,j = 0

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006), #A26 3 over ll (i, j) Z Z 2 such tht i 0, j 0, we get ( )( 2 )A 0,0 ( ) A 0,j ( 2 ) A i,0 + A i,j = 0. j Z 2 j 0 i Z i 0 (i,j) Z Z 2 i 0,j 0 But A 0,0 = nd A i,0, A 0,j 0 for ll i, j 0, so we get or, since [A =, (i,j) Z Z 2 i 0,j 0 A i,j ( )( 2 ) [A + ( [A ) 2 + [A, which is to sy tht [A ( [A ) 2, contrdiction. At this point, before moving on to prove the equivlent of Proposition 5 for n-dimensionl rrys, we note tht we cn lredy use Proposition 5 to prove Theorem for 3-dimensionl rrys. The proof is quite similr to the proof of the cse n = 3 of Theorem 5, reflecting the prllel between inequlities (7) nd (0). Proposition 6. Theorem is true for 3-dimensionl rrys. Proof. Let 2 3 nd let λ 2 = gcd(, 2 ), λ 3 = gcd(, 2, 3 ). Let A be nonnegtive integer-vlued 2 3 cyclotomic rry. Let A 0,..., A 3 be the 3-lyers of A. Let A be 3-lyer chosen such tht [A = min([a r : r Z 3 ). Put A r = A r A for ll r Z 3. Then A r is cyclotomic rry of size 2 by Proposition nd [A r 0 for ll r. If A r 0 for ll r then [A r Z + (, 2 ) for ll r by Theorem for 2-dimensionl rrys, so tht [A = [A 3 + r Z 3 [A r Z + (, 2, 3 ). We cn therefore ssume there is some s Z 3 such tht [A s > 0. Becuse [A s [A, Proposition 5 gives tht [A s ( [A ) 2. Therefore, since [A r 0 for ll r, r Z 3 [A r [A s ( [A ) 2. As the sum of the entries of ny integer-vlued cyclotomic rry is congruent to 0 modulo the gcd of the sidelengths (becuse ny integer-vlued cyclotomic rry cn be written s n integer liner combintion of fibers, cf. [9) we hve [A 0 mod λ 3 nd [A r 0 mod λ 2 for ll r. Let u 0 be the lest integer such tht [A u 3 0 mod λ 2. Note tht u λ 2 /λ 3 nd tht u [A since [A [A 3 = r Z 3 [A r 0 mod λ 2. We thus get [A u 3 = ([A u) 3 + r Z 3 [A r ([A u) 2 + ( [A ) 2

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006), #A26 4 = 2 u 2 2 (λ 2 /λ 3 ) 2 = 2 /λ 2 + ( 2 2 /λ 2 ) (λ 2 /λ 3 ) 2 2 /λ 2 + λ 2 ( 2 2 /λ 2 ) (λ 2 /λ 3 ) 2 2 /λ 2 > 2 /λ 2 2 so [A u 3 Z + (, 2 ) by Proposition 4, nd [A Z + (, 2, 3 ). We now prove inequlity (0) by generlizing the proof of Proposition 5 to n-dimensionl rrys: Lemm. (cf. [6 Thm. 4.) Let A be n integer-vlued n cyclotomic rry such tht n nd such tht [A > 0. Then [A ( [A ) 2. Proof. Assume tht counterexmple exists with [A s smll s possible. We cn gin ssume (by dding -fibers or 2-fibers to the rry) tht [A = nd tht A 0,...,0 = (mening A i,...,i n 0 for (i,..., i n ) (0,..., 0)). Put F = Z Z n nd let E F be the set of coordintes in Z Z n tht hve ll nonzero entries. Summing up the reltions over ll (i,..., i n ) E, we get (i,...,in) E F E (i,...,in) F {h: i h >0} 2 contrdiction. (i,...,in) E E E ( ) (j,...,jn) {0,} n ( ) (j,...,jn) {0,} n ( ) {h: i h>0} A i,...,i n ( ) {h: i h>0} A i,...,i n (i,...,in) F {h: i h >0} 2 j +...+j n A i j,...,i nj n = 0 j +...+j n A i j,...,i nj n = 0 A i,...,i n h n i h =0 h n i h =0 3 h n ( h ) = 0 ( h ) h n ( h ) h n ( h ) ( h ) [A + ( )( 2 ) [A + ( [A ) 2 + [A [A ( [A ) 2,

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006), #A26 5 The estimte of Lemm is enough to prove Theorem in the specil cse when the two smllest dimensions of the rry re coprime, s shown by the next two corollries lso due to Lm nd Leung. These corollries do not enter into our proof of Theorem but re interesting nonetheless since they suffice, for exmple, to prove Theorem 3 on vnishing sums of roots of unity. (We lso include them to fcilitte comprison between our results nd those of [6, which re written up in different lnguge.) Corollry. (cf. [6 Thm 4.8) Let A be non-fiber miniml n cyclotomic rry where n. Then n 3 nd [A ( 3 ) + ( )( 2 ). Proof. The fct tht n 3 is simply becuse the only miniml - nd 2-dimensionl cyclotomic rrys re fibers. Let A 0,..., A n be the n-lyers of A. If A r = 0 for some r then only one n-lyer of A is nonzero nd this n-lyer is n n cyclotomic rry, so tht the corollry is estblished by induction on the dimension of the rry. We cn therefore ssume tht none of the n-lyers of A re zero. Let A be n n-lyer of A such tht [A = min([a i : i Z n ) (by the previous remrk, [A ). Put A r = A r A for ll r Z n. Thus A r is cyclotomic rry of size n by Proposition. Since A is not n n-fiber there must be some s Z n such tht A s A. We cnnot hve A s A since A is miniml, so [A s > 0. Since [A s [A we then hve [A s ( [A ) 2 by Lemm. Now becuse [A nd n 2, we hve [A = [A r r Z n s desired. ( n )[A + [A s = ( n )[A + [A + [A s n [A + ( [A ) 2 n + ( ) 2 ( 3 ) + ( )( 2 ) Corollry 2. (cf. [6 Thm 5.2) If A is nonnegtive integer-vlued cyclotomic rry of size n where n nd where, 2 re coprime, then [A Z + (,..., n ). Proof. It suffices to consider the cse when the rry A is miniml. If A is fiber then [A = i for some i, nd the result is trivil. Otherwise n 3 nd [A ( 3 ) + ( )( 2 ) > 2 2 by Corollry, which implies [A Z + (, 2 ) by Proposition 4. From Lemm nd Proposition 4 we know tht if A is n integer-vlued 2 cyclotomic rry such tht [A > 0 nd such tht ( [A ) 2 > 2 /λ 2, where λ = gcd(, 2 ), then [A Z + (, 2 ) since [A 0 mod λ. Solving this inequlity for [A, we get: ( [A ) 2 > 2 /λ 2

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006), #A26 6 ( [A ) 2 > 2 /λ 2 [A > /λ [A /λ. Therefore [A Z + (, 2 ) for ny 2 integer-vlued cyclotomic rry A such tht [A /λ, where λ = gcd(, 2 ). This strengthening of the 2-dimensionl cse of Theorem surprisingly extends to ll dimensions. More precisely, we hve the following generliztion of Theorem : Theorem 6. If n nd A is n integer-vlued cyclotomic rry of size n such tht [A /λ where λ = gcd(,..., n ), then [A Z + (,..., n ). Theorem will be estblished s corollry of Theorem 6, whose proof is found further down. Theorem 6 is trivilly true for -dimensionl rrys nd is true for 2-dimensionl rrys by the preceding remrks. We prove the 3-dimensionl cse next. This result is lso covered by our generl proof of Theorem 6, which hndles ll rrys of dimension 3 or more, so the reder cn skip it if they wish. Proposition 7. Theorem 6 is true for n = 3. Proof. Let 2 3 nd let λ 2 = gcd(, 2 ), λ 3 = gcd(, 2, 3 ). Let A be n 2 3 integer-vlued cyclotomic rry such tht [A /λ 3. Let A 0,..., A 3 be the 3-lyers of A. Since [A < 3 there is t lest one 3-lyer of A with no negtive entries. Let A be such nonnegtive 3-lyer, chosen such tht [A = min([a r : A r 0). Put A r = A r A for ll r. We cn ssume [A < since otherwise [A [A r r Z 3 [A r [A r:a r 0 [A {r : A rn 0} [A ( 3 [A ) [A ( 3 ( /λ 3 )) ( /λ 3 ) ( 3 3 ( /λ 3 )) ( /λ 3 ) = 3 /λ 3 ( /λ 3 ) > 3 /λ 3 3 which implies by Proposition 4 tht [A Z + (, 2, 3 ). If A r 0 for ll r then [A r Z + (, 2 ) by Theorem for 2-dimensionl rrys, so tht [A = [A 3 + r Z 3 [A r Z + (, 2, 3 ).

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006), #A26 7 We cn therefore ssume there is some s Z 3 such tht [A s > 0. Using Lemm we get [A r = [A s + [A r r Z 3 r s ( [A s ) 2 + r s [A r >0 [A r ( [A s [A ) 2 + r s [A r >0 2 [A 2 2 [A s + r s [A r >0 2 [A 2 2 [A + r s [A r >0 2 [A 2 2 [A 2 [A 2 2 ( /λ 3 ) = 2 /λ 3 [A 2. (( [A r ) 2 ) (( [A r [A ) 2 ) (( [A ) 2 ) Let u 0 be the lest integer such tht [A u 3 0 mod λ 2. Note tht u λ 2 /λ 3 nd tht u [A since [A [A 3 = r [Ar 0 mod λ 2. We thus get [A u 3 = ([A u) 3 + r [A r ([A u) 2 + 2 /λ 3 [A 2 = 2 /λ 3 u 2 2 /λ 3 (λ 2 /λ 3 ) 2 = 2 /λ 2 + ( 2 /λ 3 2 /λ 2 ) (λ 2 /λ 3 ) 2 2 /λ 2 + λ 2 ( 2 /λ 3 2 /λ 2 ) (λ 2 /λ 3 ) 2 = 2 /λ 2 > 2 /λ 2 2 so [A u 3 Z + (, 2 ) by Proposition 4, nd thus [A Z + (, 2, 3 ). The proof of Theorem 6 requires number-theoretic proposition generlizing the bound u λ 2 /λ 3 found in the proofs of Propositions 6 nd 7. To better understnd the sttement, note tht if λ,,..., n re nturl numbers nd λ = gcd(λ,,..., n ) then the quntity λ/λ is equl to the number of different possible vlues tht n integer combintion of the i s cn tke mod λ. Proposition 8. Let λ,,..., n N, U,..., U n Z +. Let λ = gcd(λ,,..., n ). Then there re u,..., u n Z, 0 u i U i for i n, such tht (U u ) + +(U n u n ) n 0 mod λ nd such tht u + + u n λ/λ.

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006), #A26 8 Proof. There obviously exist integers u,..., u n such tht 0 u i U i for ll i nd such tht (U u ) + + (U n u n ) n 0 mod λ, nmely u i = U i for ll i. Now mong ll such choices of tuples (u,..., u n ) we cn ssume tht we hve chosen tuple tht minimizes the sum u + + u n. All tht we hve left to prove is tht u + + u n λ/λ. Let S 0 = U + + U n n. For t u let S t = S 0 t. For t u 2 let S u +t = S u t 2. Continue like this until S u + +u n hs been defined, which is equl to (U u ) + + (U n u n ) n. Note tht for ech 0 g < h u +... + u n we hve S g S h = n i= v i i for some v,..., v n such tht 0 v i u i for ll i. If S g S h mod λ then the tuple u = u v,..., u n = u n v n hs 0 u i U i for ll i, nd (U u ) + +(U n u n ) n 0 mod λ nd u + +u n < u + +u n, contrdiction. Therefore no two S h s hve the sme vlue mod λ. However since ech S h is n integer liner combintion of the i s there re only λ/λ different possible vlues for the S h s mod λ. It follows tht {0,, 2,..., u + + u n } λ/λ, i.e., tht u + + u n λ/λ. We still need to finlize some nottion before proving Theorem 6. If A is cyclotomic rry of size n then A r stnds for the r-th n-lyer of A, A r,t stnds for the t-th (n )-lyer of the r-th n-lyer of A, nd so forth. We sy tht r is the index of the n-lyer A r. If {r Z n : A r 0} then we define A to be nonnegtive n-lyer of A such tht [A = min([a r : A r 0, r Z n ). If severl choices re vilble for A then we cn choose the n-lyer with lest index (this convention simply llows A to be uniquely defined, nd is not otherwise importnt). If {r Z n : A r 0} = then A is undefined. Note tht A is lwys well-defined if [A < n, for then t lest one n-lyer of A is nonnegtive. If A is well-defined then we let A r = A r A for ll r Z n. Thus A r is cyclotomic rry of size n by Proposition. To prctice this nottion little more, note for exmple tht A r,s = A r,s A r, = A r,s A,s A r, provided A nd A r, re well-defined (the prerequisite for A r,s to be well-defined). On the other hnd A r, is generlly not equl to A r, A,, since the quntity represented by the rightmost vries ccording to the superscript preceding it. We will mostly be deling with rrys of the form A rn,r n,...,r k+,,r k,...,r i. It is worth emphsizing tht ny rry of this type is nonnegtive (s A r n,r n,...,r k+, is nonnegtive rry). We cn now prove Theorem 6 nd thus estblish the pper s min result. Proof of Theorem 6: Let A be n integer-vlued cyclotomic rry of size n such tht n nd [A /λ n, where λ n = gcd(,..., n ). We need to show [A Z + (,..., n ). We cn ssume tht n 3. We define sttements X i, Y i nd Z i for 3 i n by X i = A r n,r n,...,r i+, is well-defined for ll (r n, r n,..., r i+ ) Z n Z i+

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006), #A26 9 Y i = X i nd [A rn,r n,...,r i+, < for ll (r n, r n,..., r i+ ) Z n Z i+ Z i = X i nd n [Arn,...,r k+,,r k,...,r i < for ll (r n,..., r i ) Z n Z i (The vrible r k will lwys denote n index tking vlues in the set Z k, for k n. In prticulr we will use r k s shorthnd for r k Z k.) Note tht A rn,...,r k+,,r k,...,r i is well-defined whenever A rn,r n,...,r i+, is well-defined, so the sttement Z i mkes sense. The reder my check the following esy implictions: X i = A r n,r n,...,r i+,r i is well-defined for ll (r n, r n,..., r i ) Z n Z i X i Z i = X i+ for 3 i < n = (X i Y i ) for 3 i n The Theorem will follow from proving the following clims C C5: C : X n nd ((Y n Z n ) [A Z + (,..., n )) re true sttements C2 : Z i = (X i [A Z + (,..., n )) for i > 3 C3 : (Z i X i ) = (Y i [A Z + (,..., n )) for i > 3 C4 : (Z i Y i ) = (Z i [A Z + (,..., n )) for i > 3 C5 : Z 3 = [A Z + (,..., n ) Note tht clims C2 C4 imply Z i = (Z i [A Z + (,..., n )). Before proving clims C C5 we wish to mke one observtion nd prove two mini-lemms. Observtion: If 3 i n nd X i holds (i.e. if A rn,r n,...,r i is well-defined for ll (r n, r n,..., r i ) Z n Z i ) then we hve [A = [A n + r n [A rn = [A n + ( [A rn, n + ) r n r n [A r n,r n ( = [A n + [A rn, ) n + [A rn,rn r n r n,r n =... ( ) ( ) = [A n + [A rn, n + [A rn,rn, n 2 +... r n r n,r n ( + [A rn,...,ri+, ) i + [A rn,...,ri, r n,...,r i+ r n,...,r i

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006), #A26 20 which we cn rewrite more succinctly s [A = U n n + U n n + + U i i + r n,...,r i [A rn,...,ri (3) where U n = [A nd U k = r n,...,r k+ [A rn,...,rk+, for i k n. We will keep this definition of the U k s for the rest of the proof (thus U k is well-defined if nd only if X k is true). Note the U k s re nonnegtive integers becuse rrys of the type A rn,r n,...,r k+, re nonnegtive. Mini-Lemm : If A rn,...,r i is well-defined then [A rn,...,r i [A rn,...,r i + n [Arn,...,r k+,,r k,...,r i. Proof: This follows simply becuse so s climed. A r n,...,r i = A r n,...,r i+,r i A r n,...,r i+, = A rn,...,r i+2,r i+,r i A rn,...,r i+2,,r i A rn,...,r i+, =... = A rn,...,r i A r n,...,r k+,,r k,...,r i [A r n,...,r i [A rn,...,r i + [A r n,...,r k+,,r k,...,r i Mini-Lemm 2: If 3 i n nd Z i is true then [A rn,...,r i 2 [A rn,...,r i for ny subset R of Z n Z i. (r n,...,r i ) R (r n,...,r i ) R Proof: Z i implies n [Ar n,...,r k+,,r k,...,r i <, so by Mini-Lemm we hve [A rn,...,r i [A rn,...,r i + [A rn,...,r k+,,r k,...,r i [A rn,...,r i + for ll (r n,..., r i ) Z n Z i. Now since A rn,...,r i is cyclotomic rry of dimension i nd i 2 we cn pply Lemm to get: [A r n,...,r i [A r n,...,r i (r n,...,r i ) R (rn,...,r i ) R: [A rn,...,r i >0

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006), #A26 2 s climed. (rn,...,r i ) R: [A rn,...,r i >0 (rn,...,r i ) R: [A rn,...,r i >0 2 2 (( [A rn,...,r i ) 2 ) (( [A rn,...,r i + ) 2 ) r [A n,...,r i (rn,...,r i ) R: [A r n,...,r i >0 (r n,...,r i ) R [A rn,...,r i We now prove clims C-C5, from which the Theorem follows. Proof of clim C: The sttement X n is A is well-defined. However [A /λ n < n so the sttement X n is true. Becuse X n is true the sttements Y n nd Z n re both equivlent to [A <. It is thus sufficient to show ([A ) = [A Z + (,..., n ) in order to show ((Y n Z n ) [A Z + (,..., n )). Thus, ssume tht [A. Then [A = r n [A rn r n:a rn 0 [A rn [A [A {r n : A rn 0} [A ( n [A ) [A ( n ( /λ n )) ( /λ n ) ( n n ( /λ n )) ( /λ n ) = n /λ n ( /λ n ) > n /λ n n which implies by Proposition 4 (since [A 0 mod λ n ) tht [A Z + (,..., n ), s desired. This completes the proof of clim C. Proof of clim C2: We will prove (Z i X i ) = [A Z + (,..., n ). Note tht Z i = X i so A r n,r n,...,r i is well-defined for ll (r n,..., r i ) Z n Z i. Since A rn,r n,...,r i, is well-defined if [A rn,...,r i < i (s A rn,...,r i is n rry of size i ), X i implies there exists some (s n,..., s i ) Z n Z i such tht [A s n,...,s i i.

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006), #A26 22 Since i 2, Mini-Lemm 2 implies [A r n,...,r i i nd by Mini-Lemm, (rn,...,r i ) (sn,...,s i ) (rn,...,r i ) (sn,...,s i ) [A rn,...,r i (4) [A s n,...,s i [A s n,...,s i [A sn,...,s i [A sn,...,s i [A s n,...,s k+,,s k,...,s i U k. (5) Let λ i = gcd(,..., i ) for i n (λ n ws lredy defined like this). Then r n,...,r i [A rn,...,r i 0 mod λ i since ech A rn,...,r i is n i cyclotomic rry. It follows from (3) nd Proposition 8 tht there exist nonnegtive integers u i,..., u n with u k U k nd u i + + u n λ i /λ n such tht [A u n n u i i 0 mod λ i. Since [A s n,...,s i i we hve by Mini-Lemm tht so [A sn,...,s i + U k Therefore, using (3), (4) nd (5), = = [A u k k [A s n,...,s k+,,s k,...,s i [A sn,...,s k+,,s k,...,s i (U k u k ) k + [A s n,...,s i + (U k u k ) i [A sn,...,s i ( [A sn,...,s i. (rn,...,r i ) (sn,...,s i ) [A r n,...,r i U k i U k ) ( i ) [A sn,...,s i i (rn,...,r i ) (sn,...,s i ) (rn,...,r i ) (sn,...,s i ) ( [A sn,...,s i )( i ) [A sn,...,s i i [A rn,...,r i [A rn,...,r i (rn,...,r i ) (sn,...,s i ) u k i [A rn,...,r i u k i

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006), #A26 23 = ( i ) i [A u k i ( i ) i ( /λ n ) (λ i /λ n ) i (6) = i /λ i + ( i /λ n i /λ i ) (λ i /λ n ) i (7) i /λ i + λ i ( i /λ n i /λ i ) (λ i /λ n ) i (8) = i /λ i (9) > i /λ i i (20) which mens [A n u k k Z + (,..., i ) by Proposition 4 nd so [A Z + (,..., n ), s desired. This concludes the proof of clim C2. Proof of clim C3: We will prove tht (Z i X i Y i ) = [A Z + (,..., n ). Since Y i = X i nd [A rn,...,r i, < for ll (r n,..., r i ) Z n Z i, (X i Y i ) implies there exists (s n,..., s i ) Z n Z i such tht [A s n,...,s i,. By the proof of clim C2 we cn ssume tht [A rn,...,r i < i for ll (r n,..., r i ) Z n Z i. If i = then = 2 =... = i so [A rn,...,r i 0 mod i for ll (r n,..., r i ) Z n Z i. But since [A rn,...,r i [A rn,...,r i > i we then hve [A rn,...,r i Z + ( i ) so [A Z + (,..., n ) by (3). We cn therefore ssume i >. Mini-Lemm 2 implies tht [A r n,...,r i i (rn,...,r i ) (sn,...,s i ) On the other hnd, since i > we now hve [A sn,...,s i = r i [A sn,...,s i,r i r i :A sn,...,s i,r i 0 Combining (2) nd (22) we obtin (rn,...,r i ) (sn,...,s i ) [A sn,...,s i,r i [A sn,...,s i [A sn,...,s i, {r i : A sn,...,s i,r i 0} [A sn,...,s i ( i [A sn,...,s i ) [A sn,...,s i i i [A sn,...,s i i i ([A sn,...,s i + [A rn,...,r i. (2) ) [A sn,...,s k+,,s k,...,s i. (22) r n,...,r i [A rn,...,ri i i [A i [A sn,...,s k+,,s k,...,s i. (23)

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006), #A26 24 Tke u i,..., u n s in the proof of clim C2. By (23) nd becuse U k = r n,...,r k+ [A rn,...,r k+, [A sn,...,s k+,,s k,...,s i we get = = [A u k k (U k u k ) k + [A rn,...,ri r n,...,r i (U k u k ) i + i i [A i [A sn,...,s k+,,s k,...,s i (U k [A sn,...,s k+,,s k,...,s i ) i u k i + i i [A (λ i /λ n ) i + i i ( /λ n ) = i /λ i + ( i /λ n i /λ i ) (λ i /λ n ) i i /λ i + λ i ( i /λ n i /λ i ) (λ i /λ n ) i = i /λ i > i /λ i i so [A n u k k Z + (,..., i ) by Proposition 4 nd [A Z + (,..., n ), s desired. This concludes the proof of clim C3. Proof of clim C4: We prove (Z i Y i Z i ) = [A Z + (,..., n ). If Z i then there exists some (s n,..., s i ) Z Z i such tht n [As n,...,s k+,,s k,...,s i. Let A = {r i : n [Asn,...,s k+,,s k,...,s i,r i } (we know A since s i A). We hve [A sn,...,s i,r i r i / A r i / A: [A sn,...,s i,r i >0 r i / A: [A s n,...,s i,r i >0 r i / A: [A sn,...,s i,r i >0 2 r i / A: [A sn,...,s i,r i >0 [A sn,...,s i,r i (( [A sn,...,s i,r i ) 2 ) ( ( [A sn,...,s i,r i [A sn,...,s i,r i ) [A sn,...,s k+,,s k,...,s i,r i ) 2

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006), #A26 25 2 [A sn,...,s i,r i r i / A nd r i A r i A r i A [A s n,...,s i,r i [A sn,...,s i,r i ( [A sn,...,s i,r i ) [A sn,...,s k+,,s k,...,s i,r i [A s n,...,s k+,,s k,...,s i [A s n,...,s i, A [A sn,...,s i,r i. r i A Therefore [A sn,...,si,ri r i [A sn,...,s k+,,s k,...,s i [A sn,...,si, A 2 [A sn,...,s i so we get [A s n,...,s i = [A s n,...,s i, i + r i [A s n,...,s i,r i [A s n,...,s i, i 2 [A sn,...,s i [A s n,...,s k+,,s k,...,s i [A s n,...,s i, A (24) On the other hnd, since we re ssuming Z i nd since i 2, Mini-Lemm 2 implies tht [A rn,...,r i i [A rn,...,r i. (25) (rn,...,r i ) (sn,...,s i ) Combining (24) nd (25) gives us r n,...,r i [A rn,...,ri [A sn,...,si, i (rn,...,r i ) (sn,...,s i ) [A s n,...,s k+,,s k,...,s i [A sn,...,s i, A i [A. Since we know [A sn,...,si, < (from Y i ) we know tht r i A = [A sn,...,s k+,,s k,...,s i,r i,

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006), #A26 26 therefore r i s i [A sn,...,s k+,,s k,...,s i,r i A nd [A s n,...,s k+, [A s n,...,s i, + [A s n,...,s k+,,s k,...,s i = [A sn,...,si, + [A sn,...,s k+,,s k,...,s i,r i r i + [A sn,...,s k+,,s k,...,s i,r i r i s i + A. Becuse U k = r n,...,r k+ [A r n,...,r k+, [A s n,...,s k+, we then obtin U k + [A sn,...,si, + A. Let u i,..., u n be defined s in the proofs of clims C2 nd C3. We now hve tht = [A u k k (U k u k ) k + [A rn,...,ri r n,...,r i (U k u k ) i + [A sn,...,si, i [A sn,...,s k+,,s k,...,s i [A s n,...,s i, A i [A U k ( i ) + [A sn,...,si, ( i ) [A sn,...,si, ( A ) i [A u k i ( + A )( i ) [A sn,...,si, ( A ) i [A ( i ) i [A u k i ( i ) i ( /λ n ) (λ i /λ n ) i u k i

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006), #A26 27 > i /λ i i (where the lst inequlity is obtined s in (6)-(20)), so [A n u k k Z + (,..., i ) nd [A Z + (,..., n ), s desired. This concludes the proof of clim C4. Proof of clim C5: Assume Z 3. Then A rn,...,r 3 is well-defined for ll (r n,..., r 3 ) Z n Z 3 nd n k=3 [Ar n,...,r k+,,r k,...,r 3 < for ll (r n,..., r 3 ) Z n Z 3. Note tht A rn,...,r 3 is cyclotomic rry of size 2. If A rn,...,r 3 0 for ll (r n,..., r 3 ) Z n Z 3 then [A r n,...,r 3 Z + (, 2 ) by Proposition 2 so [A Z + (,..., n ) by (3). Therefore we cn ssume there is some (s n,..., s 3 ) Z n Z 3 such tht [A sn,...,s 3 > 0. We hve tht (rn,...,r 3 ) (sn,...,s 3 ) [A rn,...,r 3 2 (rn,...,r 3 ) (sn,...,s 3 ) [A rn,...,r 3 nd tht So [A sn,...,s 3 ( [A sn,...,s 3 ) 2 ( ) [A sn,...,s 3 [A sn,...,s k+,,s k,...,s 3 2 k=3 = 2 2 [A sn,...,s k+,,s k,...,s 3 2 [A sn,...,s 3. k=3 r n,...,r 3 [A rn,...,r3 2 2 [A s n,...,s k+,,s k,...,s 3 2 [A. (26) k=3 Becuse X 3 is true, [A = U n n + + U 3 3 + r n,...,r 3 [A rn,...,r3 where r n,...,r 3 [A rn,...,r 3 0 mod λ 2. By Proposition 8 there exist integers u 3,..., u n such tht (i) 0 u k U k for ll 3 k n, (ii) u 3 + +u n λ 2 /λ n, nd (iii) [A n k=3 u k k 0 mod λ 2. Using (26) we get = [A u k k k=3 (U k u k ) k + [A rn,...,r3 r n,...,r 3 (U k u k ) 2 + 2 2 [A sn,...,s k+,,s k,...,s 3 2 [A k=3 k=3 k=3

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006), #A26 28 = (U k [A s n,...,s k+,,s k,...,s 3 ) 2 + 2 2 [A 2 k=3 2 2 [A 2 k=3 2 2 ( /λ n ) 2 (λ 2 /λ n ) = 2 /λ 2 + ( 2 /λ n 2 /λ 2 ) 2 (λ 2 /λ n ) 2 /λ 2 + λ 2 ( 2 /λ n 2 /λ 2 ) 2 (λ 2 /λ n ) = 2 /λ 2 > 2 /λ 2 2 u k which shows [A n k=3 u k k Z + (, 2 ) nd thus [A Z + (,..., n ), s desired. This concludes the proof of clim C5 nd the proof of Theorem 6. k=3 u k References [ A. Bruer, On problem of prtitions, Amer. J. Mth. 64 (942), 299-32. [2 A. Bruer nd J. E. Shockley, On problem of Frobenius, J. Reine Angew. Mth. 2 (962), 25-220. [3 J.H. Conwy nd A.J. Jones, Trigonometric diophntine equtions (On vnishing sums of roots of unity), Act Arithmetic, 30 (976), 229-240. [4 P. Erdős nd R. L. Grhm, On liner Diophntine problem of Frobenius, Act Arith., 2 (972), 399-408. [5 M. Hertweck, Contributions to the integrl representtion theory of groups, Hbilittionsschrift, University of Stuttgrt (2004). [6 T.Y. Lm nd K.H. Leung, On vnishing sums of roots of unity, J. Algebr, 224 no. (2000), 9-09. [7 H. B. Mnn, On liner reltions between roots of unity, Mthemtik, 2 Prt 2 no. 24 (965), 07-7. [8 B. Poonen nd M. Rubinstein, The number of intersection points mde by the digonls of regulr polygon, SIAM J. on Disc. Mth., no. (998), 33-56. [9 J.P. Steinberger, Miniml vnishing sums of roots of unity with lrge coefficients, preprint. [0 J.P. Steinberger, Indecomposble tilings of the integers with exponentilly long periods, Electronic Journl of Combintorics, 2 (2005).