Expected Norms of Zero-One Polynomials

Transcription

1 DRAFT: Caad. Math. Bull. July 4, 08 :5 File: borwei80 pp. Page Sheet of Caad. Math. Bull. Vol. XX (Y, ZZZZ pp. 0 0 Expected Norms of Zero-Oe Polyomials Peter Borwei, Kwok-Kwog Stephe Choi, ad Idris Mercer 5 5 Abstract. Let A = a 0 + a z + + a z : a j 0, }, whose elemets are called zerooe polyomials ad correspod aturally to the subsets of [] := 0,,..., }. We also let A,m = α(z A : α( = m}, whose elemets correspod to the ` m subsets of [] of size m, ad let B = A + \ A, whose elemets are the zero-oe polyomials of degree exactly. May researchers have studied orms of polyomials with restricted coefficiets. Usig α p to deote the usual L p orm of α o the uit circle, oe easily sees that α(z = a 0 +a z+ +a N z N R[z] satisfies α = c 0 ad α = c 0 + (c + + c N, where c k := P j=0 a ja j+k for 0 k N. If α(z A,m, say α(z = z β + + z β m where β < < β m, the c k is the umber of times k appears as a differece β i β j. The coditio that α A,m satisfies c k 0, } for k is thus equivalet to the coditio that β,..., β m } is a Sido set (meaig all differeces of pairs of elemets are distict. 6 I this paper, we fid the average of α over α A, α B, ad α A,m. We further 6 show that our expressio for the average of α over A,m yields a ew proof of the kow result: if 8 m = o( /4 ad B(, m deotes the umber of Sido sets of size m i [], the almost all subsets 8 of [] of size m are Sido, i the sese that lim B(, m/` m =. Itroductio ad Statemet of Mai Result 3 3 We let A deote the set a 0 + a z + + a z : a j 0, } for all j}, ad we call the elemets of A zero-oe polyomials. There is a atural bijectio betwee the 35 polyomials i A ad the subsets of [] := 0,,..., }. Geerally, if α(z A, we defie m := α( = the umber of coefficiets of α(z that are, ad we write α(z = z β + z β + + z β m where β < β < < β m, so 40 4 β 4, β,...,β m } is the subset of [] that correspods to α(z. We let A,m deote 4 the set α(z A 4 : α( = m}, so A,m = ( m ad A = A,0 A, A,. 43 We also defie B 43 := A + \ A, so B cosists of the zero-oe polyomials of degree exactly. A recurrig theme i the literature is the problem of fidig a polyomial with small orm subject to some restrictio o its coefficiets. (See [3, Problem 6], 4 [5, Problem ], or [, Ch. 4, 5]. I geeral, for 4 (. α(z = a a z + + a N z N R[z], 50 5 Received by the editors May, The research of Peter Borwei ad Stephe Choi is supported by NSERC of Caada AMS subject classificatio: Primary: B83; secodary: C08, C0. 53 c Caadia Mathematical Society ZZZZ

2 DRAFT: Caad. Math. Bull. July 4, 08 :5 File: borwei80 pp. Page Sheet of P. Borwei, K.-K. S. Choi, ad I. Mercer we defie the usual L p orms of α(z o the uit circle: 0 ( 0 π α p := α(e iθ /p, dθ p π 0 where p is real. The mai result of this paper, which appears as Theorem 4. i Sectio 4, is that if 4 ad m, we have 5 5 E A ( α = (, 6 E A,m ( α = m m + m[4] 3( 3 + m[3] ( m( 4 + (, E B ( α = (, 6 where E Ω ( α deotes the average of α over the polyomials i Ω, ad the otatio x [k] is shorthad for x(x (x k +. This complemets results of 6 Newma ad Byres [], who foud the average of α over the polyomials of 6 the form 8 8 (. a 0 + a z + + a z, a j +, } for all j, ad Borwei ad Choi [], who foud (amog other thigs the average of α 6 6 ad α over the polyomials of the form (., ad the average of α, α, 3 ad α over the 3 polyomials of the form a a z + + a z, a j +, 0, } for all j Autocorrelatio Notice that if α is of the form (. ad z =, we have ( 40 4 α(z = α(zα(z = (a 0 + a z + + a N z N a 0 + a 4 z + + a N z N = c N 43 z z + c 0 + c z + + c N z N, where the c k are the so-called (aperiodic autocorrelatios of α, defied for 0 k N by c k := j=0 a ja j+k. Usig the geeral fact that π ( 4 b M 4 π 0 z M + + b z + b 0 + b z + + b M z M dθ = b 0, (z = e iθ, 5 we see that for α of the form (., we have α 53 = π ( c N π 0 z N + + c z + c 0 + c z + + c N z N dθ = c 0, (z = e iθ, 55

3 DRAFT: Caad. Math. Bull. July 4, 08 :5 File: borwei80 pp. Page 3 Sheet 3 of Expected Norms of Zero-Oe Polyomials 3 ad 0 (. 0 α = π ( c N π 0 z N + + c z + c 0 + c z + + c N z N dθ = cn + + c + c0 + c + + cn = c0 + (c + + cn, (z = e iθ. 5 5 We further observe that, say, ( c k = a j a j+k = a i a i+k a j a j+k = a i a j a i+k a j+k j=0 i=0 j=0 i=0 j=0 = f (i, j. i=0 j=0 Notig that f (i, j := a i a j a i+k a j+k satisfies f (i, j = f ( j, i, we have 6 6 (. ck = f (i, j = f (i, i + f (i, j 8 8 i=0 j=0 i=0 0 i< j = a i a i+k + a i a j a i+k a j+k. 3 i=0 0 i< j 3 If α(z = a a z = z β + + z β m A,m, the we have c 0 = m ad c k is 35 the umber of j such that a j ad a j+k are both ad is equal to the umber of times k appears as a differece β i β j. Thus c + + c = m(m /, ad sice the Author: Redudat (? displayed lies deleted c k are oegative itegers, we have (.3 c c c + + c = m(m / 3 with equality if ad oly if c k 0, } for k, or i other words, if ad oly 4 4 if all differeces of pairs of elemets of β,...,β m } are distict. If all differeces of 4 4 pairs of elemets of β,...,β m } are distict, we call β,...,β m } a Sido set. Usig (., we see that (.3 ad c 0 = m prove the followig. Propositio. For ay α(z = z β + + z β m A,m, we have α m m, with equality if ad oly if β,...,β m } is a Sido set. We observe also that (.3 implies that c + +c m(m / is a oegative iteger, ad is zero if ad oly if β 4,...,β m } is Sido. 4 3 Some Facts ad Notatio If Ω deotes A 53, B, or A,m, the we tur Ω ito a probability space by givig each 53 polyomial α Ω equal weight p(α. 55

4 DRAFT: Caad. Math. Bull. July 4, 08 :5 File: borwei80 pp. Page 4 Sheet 4 of 4 P. Borwei, K.-K. S. Choi, ad I. Mercer Geerally, we will deote a typical elemet of A or A,m by 0 0 α(z = a 0 + a z + + a z, ad deote a typical elemet of B by α(z = a 0 + a z + + a z + z. As i Sectio, if α A,m, we also write 5 5 α(z = z β + z β + + z β m where β < β < < β m. If Ω is oe of the three spaces A, B, or A,m ad X is a radom variable o Ω, we of course have E Ω (X = α Ω X(αp(α, ad we will sometimes omit the subscript Ω if it is clear from the cotext what probability space we are cosiderig. Two facts we will use that are each immediate from first priciples are Markov s iequality, Pr[X a] E(X/a, where X is a oegative real radom variable, ad liearity of expectatio, E(X + +X k = E(X + +E(X k, which holds regardless of depedece or idepedece of the X i Calculatio of E( α 8 8 Let j, j, j 3, j 4 deote distict itegers. We begi this sectio by fidig some averages of products of a ji that we will eed later. First, suppose our probability space Ω is A. We the have 3 3 (4. E(a j a j = (umber of α A such that a j = a j = = = 4, 36 ad by similar reasoig, we have (4. E(a j a j a j3 = /8, E(a j a j a j3 a j4 = / Now suppose our probability space Ω is A,m. We the have (4.3 E(a j a j = ( (umber of α A,m such that a j = a j = 43 m ( m m(m = ( = ( = m[] m ad by similar reasoig, we have (4.4 E(a 5 j a j a j3 = m [3] / [3], E(a j a j a j3 a j4 = m [4] / [4] We ote that we eed 4 i order for all expressios i (4.3 ad (4.4 to be defied. For Ω = A,m, the case 3 will be treated separately. 55

5 DRAFT: Caad. Math. Bull. July 4, 08 :5 File: borwei80 pp. Page 5 Sheet 5 of Expected Norms of Zero-Oe Polyomials 5 Now if Ω is either of the probability spaces A or A,m, the equatio (. gives 0 0 k (4.5 ck = a i a i+k + a i a i+k a j a j+k. i=0 0 i< j k We defie λ := k ad also defie 5 5 λ (4.6 S := a i a i+k, i=0 (4. T := a i a j a i+k a j+k, 0 i< j λ which of course implies ck = S + T. If k = 0, the k = m. So if Ω = A,m, we have simply E(c0 = m, whereas if Ω = A, we have ( 6 (4.8 E(c0 = m 6. m=0 8 8 It is a exercise to see that the right side of (4.8 evaluates to ( + /4. Alteratively, we may observe that c 0 has a biomial distributio with parameters ad /, which implies 3 3 (4. E(c 0 = Var(c 0 + E(c 0 = ( + + =. 4 Havig foud E(c 36 0 for Ω = A,m ad for Ω = A, we ow shift our attetio 36 to E(c 3 k for k 0. 3 Assume k 0, ad observe that (4.5, (4.6, ad (4. (ad liearity of expectatio give us λ 4 (4.0 E(c 4 k = E(S + E(T = E(a i a i+k + E(a i a j a i+k a j+k. 4 i=0 0 i< j λ 4 Sice k 0, each of the λ terms i the sum E(S is of the form E(a j a j where j j. We thus have 4 λ/4 if Ω = A 4, (4. E(S = λm [] / [] if Ω = A,m, 5 by (4. ad (4.. As for the ( λ terms i the sum E(T, each term is of the form 5 5 E(a i a j a i+k a j+k. Sice k 0 ad i < j, the four subscripts i, j, i + k, j + k costitute 5 either three distict itegers (if j = i + k or four distict itegers (if j i + k. If i, j, i + k, j + k} cosists of three distict itegers j, j, j 3 where j 3 is the oe that is 55

6 DRAFT: Caad. Math. Bull. July 4, 08 :5 File: borwei80 pp. Page 6 Sheet 6 of 6 P. Borwei, K.-K. S. Choi, ad I. Mercer repeated, the, sice a j 0, } for all j, we have E(a i a j a i+k a j+k = E(a j a j a j 3 = E(a 0 j a j a j3, whereas, of course, if i, j, i + k, j + k} cosists of four distict itegers, 0 the E(a i a j a i+k a j+k is of the form E(a j a j a j3 a j4. Therefore, we ow ask the questio: For which of the ( λ terms i the sum E(T does the set i, j, i +k, j +k} cosist of oly three distict itegers? For some i 0,,..., λ }, there is exactly oe j satisfyig both i < j λ 5 ad j = i + k, ad for other values of i, there is o such j. We will say that i is of 5 type I if the former criterio holds, ad is of type II if the latter criterio holds. A iteger i is of type I if ad oly if i + k < λ, or equivaletly, i < λ k = k. If k 0 (i.e., if k /, the i < k ever happes, i.e., o i is of type I ad so all of the ( λ terms i the sum E(T are of the form E(a j a j a j3 a j4. O the other had, if k > 0 (i.e., if k < /, the i < k = λ k sometimes happes; amely, it happes if ad oly if i is oe of the λ k itegers 0,,...,λ k. I that case, each of those λ k values of i is of type I, which implies that precisely λ k of the ( λ terms i the sum E(T are of the form E(a j a j a j3 ad the remaiig terms are of the form E(a j a j a j3 a j4. This implies that we have 6 ( λ 6 E(a j a j a j3 a j4 if k /, ( 8 E(T = λ E(a j a j a j3 a j4 8 +(λ k [ E(a j a j a j3 E(a j a j a j3 a j4 ] if k < /. Thus, if Ω = A, the 3 ( λ 3 / if k /, E(T = ( λ / + (λ k/ if k < /, 36 ad hece by (4.0 ad (4., λ/4 + λ(λ / if k /, 38 E(c 38 k = 3 λ/4 + λ(λ / + (λ k/ if k < /. 3 4 O the other had, if Ω = A,m, the 4 4 ( λ 4 43 m [4] / [4] if k /, E(T = ( 43 λ m [4] / [4] + (λ k[m [3] / [3] m [4] / [4] ] if k < /, ad hece 4 λ m [] 4 E(c + λ(λ m[4] if k /, k = [4] 4 λ m[] + λ(λ m[4] + (λ k [ ] m [3] m[4] if k < /. [] [4] [3] [4] 4 It the follows that if Ω = A, we have ( 5 λ ( λ(λ / ( (λ k 53 (4. E(c + + c = + +, 53 4 k= k= k= 55

7 DRAFT: Caad. Math. Bull. July 4, 08 :5 File: borwei80 pp. Page Sheet of Expected Norms of Zero-Oe Polyomials whereas if Ω = A,m, we have 0 0 (4. E(c + + c = (λ m[] [] + (λ(λ m[4] [4] k= k= / ( [ 5 m [3] ] + (λ k 5 [3] m[4] [4]. k= Recallig that λ is simply shorthad for k, it is straightforward to verify that ( λ =, (λ ( ( λ =, 3 k= k= ad that / ( / if is eve, (λ k = ( / if is odd. 6 k= 6 So, if Ω = A, the from (4. we get 8 8 E(c + + c 4 = + ( ( 3 + ( if is eve, 4 ( + ( ( 3 + ( if is odd, 3 ( 3 + /6 if is eve, 3 = ( /6 if is odd, which, usig (. ad (4., implies E( α = = + if is eve, = + +3 if is odd, 3 4 or equivaletly (4. E A ( α 3 4 = (. 6 O the other had, if Ω = A,m, the from (4. we get 4 E(c c m [] 4 ( 4 = [] + m[4] ( ( [4] 3 + ( m [3] m[4] [3] ( [4] if is eve, 50 m [] ( m[4] ( ( [4] 3 + ( m [3] m[4] [3] ( [4] if is odd, 5 ( 5 m 5 = + m [4] /(3( 3 + m [3] ( m( /( [4] if is eve, ( 53 m 53 + m [4] /(3( 3 + m [3] ( m( + /( [4] if is odd, 55

8 DRAFT: Caad. Math. Bull. July 4, 08 :5 File: borwei80 pp. Page 8 Sheet 8 of 8 P. Borwei, K.-K. S. Choi, ad I. Mercer which, usig (., implies 0 m 0 E( α 4 4 = m + m [4] 3( 3 + m[3] ( m( if is eve, [4] m m + m[4] 3( 3 + m[3] ( m( + if is odd, [4] or equivaletly 5 (4.5 E A,m ( α = m m + m[4] 3( 3 + m[3] ( m( 4 + ( [4]. Notice that if m is fixed ad approaches ifiity, the E A,m ( α approaches m m, i.e., for fixed m ad large, we expect a radom α A,m to correspod to a Sido set, as is cosistet with ituitio. If Ω = B, sice B := A + \ A, we get E B ( α 4 4 = α = E A + ( α E A ( α α B = ( by (4.. Therefore we have proved 8 Theorem 4. If m, we have 8 E A ( α 4 4 = (, E A,m ( α = m m + m[4] 3( 3 + m[3] ( m( 4 + ( [4] (if 4, 36 E B ( α = ( For completeess, we also determie E A,m ( α whe 3. If 3, we have 38 3 α(z = a 0 + a z + a z ad the 3 α 4 = c 0 + c + c 4 4 = (a 0 + a + a + (a 0 a + a a + (a 0 a 4 = a a 4 + a 4 + 4(a 0a + a 0a + a a + 4a 0 a a = a 0 + a + a + 4(a 0 a + a 0 a + a a + 4a 0 a a, 4 sice a j 0, } from which it readily follows that 4 E A,0 ( α = E A 3,0 ( α = 0, 50 E A, ( α = E A 3, ( α =, E A, ( α = E A 3, ( α = 6, E A3,3 ( α =. 55

9 DRAFT: Caad. Math. Bull. July 4, 08 :5 File: borwei80 pp. Page Sheet of Expected Norms of Zero-Oe Polyomials We remark that substitutig m 0,,, 3} ito the secod equatio i Theorem 4. ad the formally cacellig commo factors as appropriate, we get 0 0 E A,3 ( α = 5 + 3( 4 + (, ( ( E A, ( α = 6, 5 5 E A, ( α =, E A,0 ( α = 0, yieldig results cosistet with the explicit averages just obtaied for 3. 5 Ubiquity of Sido Sets We show that our expressio for E A,m ( α yields a ew proof of a result that appears i articles by Godbole et al. [4] ad Nathaso [6]. Suppose Ω = A,m, ad as before, deote a typical elemet of A,m by 6 6 α(z = z β + + z β m. 8 Recall from Sectio that X := c c ( m is a oegative-iteger-valued radom variable o Ω that attais the value 0 if ad oly if β,...,β m } is a Sido set. We have 3 ( 3 m E A,m (X = E A,m (c + + c 35 m [4] 35 3( 3 36 = ( m( if is eve, [4] m [4] 36 ( 3 + m[3] ( m( + if is odd [4] m[4] 3 3( 3 + m[3] ( m( [4] 4 m(m (m (m 3 m = 4 4 6( 4 m4 3 if 4. O the other had, if we let B(, m be the umber of Sido sets i [] with 4 m elemets, the we have 4 E(X = ( 4 X = ( X ( 4 m α A,m m #α A,m : X(α > 0} α A,m,X>0 m 5 5 ( 5 B(, m. 5 m Hece we have proved (by essetially usig Markov s iequality the followig. 55

10 DRAFT: Caad. Math. Bull. July 4, 08 :5 File: borwei80 pp. Page 0 Sheet 0 of 0 P. Borwei, K.-K. S. Choi, ad I. Mercer Corollary 5. For 4 m, we have 0 ( 0 ( B(, m m4 m 3 ad 5 Pr[β,...,β m } is Sido] > m Hece if m = o( /4, the as approaches ifiity, the probability that a radomly chose m-subset of [] is Sido approaches. Although whe m = o( /4, the probability that a radomly chose m-subset of [] is Sido approaches (i.e., α is m m for almost all α A,m, there are some other cases i which a positive proportio of polyomials i A,m have very large L 4 orm. For α A,m, sice for 0 k, c k = k j=0 a j a j+k, we have c 0 = m, ad for k, c k mim, k}. Therefore, we have 6 6 m+ 8 α = c0 + ck m + (m + ( k 8 k= k= k= m+ = m 4 3 m3 + 4m 4m m 3 3 = ( + o(m ( 3 m if = o(m as m, ad o the other had, from (4.5 we have ( + o(m 4 38 E A,m ( α ( α m α A,m 3 4 = ( 4 α 4 m 4 + } α α 4 x α >x x + ( α m α >x 4 x + ( 4 ( ( + o(m 3 m. m α It the follows that for ay x < ( + o(m 4 /(3, we have #α A 53,m : α 4 ( 4 > x} 53 ( + o(m4 /(3 x ( + o(m m ( m/3. 55

11 DRAFT: Caad. Math. Bull. July 4, 08 :5 File: borwei80 pp. Page Sheet of Expected Norms of Zero-Oe Polyomials I particular, for ay ǫ > 0, if m = c ad x = c m for 0 < c < ad 0 < c 0 < ( ǫc/3, we have 0 #α A,m : α > c m } ( ǫc /3 c m ( + ǫ( c /3 > 0 5 for sufficietly large ad m, i.e., there is a positive proportio of polyomials i 5 A,m havig large L 4 orm (ote that the L 4 orm i A,m is at most as large as ( + o(m. Refereces [] P. B. Borwei, Computatioal Excursios i Aalysis ad Number Theory. CMS Books i Mathematics/Ouvrages de Mathmatiques de la SMC 0. Spriger-Verlag, New York, 0. [] P. B. Borwei ad K.-K. S. Choi. The average orm of polyomials of fixed height. To appear. Author: re: [] please furish [3] P. Erdős, Some usolved problems. Michiga Math. J. 4(5, 0. details: ame of joural to which submitted or url; or [4] A. P. Godbole, S. Jaso, N. W. Locatore, Jr., ad R. Rapoport, Radom Sido sequeces. delete. J. Number Theory 5(, o.,. [5] J. E. Littlewood, Some Problems i Real ad Complex Aalysis. D.C. Heath, Lexigto, MA, [6] M. B. Nathaso. O the ubiquity of Sido sets. I: Number Theory. Spriger, New York, 04, 6 pp. 63. [] D. J. Newma ad J. S. Byres, The L 4 orm of a polyomial with coefficiets ±. Amer. Math. 8 Mothly (0, 4. 8 Departmet of Mathematics, Simo Fraser Uiversity, Buraby, BC, V5A S6 pborwei@cecm.sfu.ca 3 kkchoi@cecm.sfu.ca 3 Departmet of Mathematics, Atkiso Faculty, York Uiversity, Toroto, ON, M3J P3 idmercer@yorku.ca