LITTLEWOOD-TYPE PROBLEMS ON SUBARCS OF THE UNIT CIRCLE. Peter Borwein and Tamás Erdélyi

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1 LITTLEWOOD-TYPE PROBLEMS ON SUBARCS OF THE UNIT CIRCLE Peter Borwein nd Tmás Erdélyi Abstrct. The results of this pper show tht mny types of polynomils cnnot be smll on subrcs of the unit circle in the complex plne. A typicl result of the pper is the following. Let F n denote the set of polynomils of degree t most n with coefficients from {, 0, }. There re bsolute constnts c > 0, c 2 > 0, nd c 3 > 0 such tht exp c / inf p L A, 0 =p F n inf p A exp c 2 / 0 =p F n for every subrc A of the unit circle D := {z C : z = } with length 0 < < c 3. The lower bound results extend to the clss of f of the form fz = nx j z j, j C, j M, m = j=m with vrying nonnegtive integers m n. It is shown tht functions f of the bove form cnnot be rbitrrily smll uniformly on subrcs of the circle. However, this does not extend to sets of positive mesure. It shown tht it is possible to find polynomil of the bove form tht is rbitrrily smll on s much of the boundry in the sense of liner Lebesgue mesure s one likes. An esy to formulte corollry of the results of this pper is the following. Corollry. Let A be subrc of the unit circle with length la =. If p k is sequence of monic polynomils tht tends to 0 in L A, then the sequence Hp k of heights tends to. The results of this pper re deling with extensions of clsses much studied by Littlewood nd mny others in regrds to the vrious conjectures of Littlewood concerning growth nd fltness of unimodulr polynomils on the unit circle D. Hence the title of the pper. 99 Mthemtics Subject Clssifiction. J54, B83. Key words nd phrses. Trnsfinite dimeter; integers; diophntine pproximtion; Chebyshev; polynomil; restricted coefficients;,0, coefficients; 0, coefficients. Reserch is supported, in prt, by the Ntionl Science Foundtion of the USA under Grnt No. DMS nd conducted while n Interntionl Postdoctorl Fellow of the Dnish Reserch Council t University of Copenhgen T. Erdélyi. Reserch is supported, in prt, by NSERC of Cnd P. Borwein. Typeset by AMS-TEX

2 2 PETER BORWEIN AND TAMÁS ERDÉLYI. Introduction Littlewood s well-known nd now resolved conjecture of round 948 concerns polynomils of the form n pz := j z kj, j= where the coefficients j re complex numbers of modulus t lest nd the exponents k j re distinct non-negtive integers. It sttes tht such polynomils hve L norms on the unit circle D := {z C : z = } tht grow t lest like c log n with n bsolute constnt c > 0. This ws proved by Konjgin [Ko-8] nd independently by McGehee, Pigno, nd Smith [MPS-8]. Pichorides, who contributed essentilly to the proof of the Littlewood conjecture, observed in [Pi-83] tht the originl Littlewood conjecture when ll the coefficients re from {0, } would follow from result on the L norm of such polynomils on sets E D of mesure π. Nmely if E n z kj dz c for ny subset E D of mesure π with n bsolute constnt c > 0, then the originl Littlewood conjecture holds. Throughout the pper the mesure of set E D is the liner Lebesgue mesure of the set {t [ π, π : e it E}. Konjgin [Ko-96] gives lovely probbilistic proof tht this hypothesis fils. He does however conjecture the following: for ny fixed set E D of positive mesure there exists constnt c = ce > 0 depending only on E such tht E n z kj dz ce. In other words the sets E ǫ D of mesure π in his exmple where must vry with ǫ > 0. E ǫ n z kj dz < ǫ We show, mong other things, tht Konjgin s conjecture holds on subrcs of the unit circle D. Additionl mteril on Littlewood s conjecture nd relted problems concerning the growth of polynomils with unimodulr coefficients in vrious norms on the

3 LITTLEWOOD-TYPE PROBLEMS ON SUBARCS OF THE UNIT CIRCLE 3 unit disk is to be found, for exmple, in [Bou-86], [Be-95], [K-85], [Li-86], [M-63], [Ne-90], [Od-93], nd [So-95]. All the results of this pper concern how smll polynomils of the bove nd relted forms cn be in the L p norms on subrcs of the unit disk. For p the results re shrp, t lest up to constnt in the exponent. An interesting relted result is due to Nzrov [N-93]. One of its simpler versions sttes tht there is n bsolute constnt c > 0 such tht n c mi mx pz mx z I ma pz z A for every polynomil p of the form pz = n jz kj with k j N nd j C nd for every A I, where I is subrc of D with length mi nd A is mesurble with Lebesgue mesure ma. This extends result of Turán [Tu-84] clled Turán s Lemm, where I = D nd A is subrc. 2. Nottion For M > 0 nd µ 0, let S µ M denote the collection of ll nlytic functions f on the open unit disk D := {z C : z < } tht stisfy fz M z µ, z D. We define the following subsets of S. Let n F n := f : fx = j x j, j {, 0, } nd denote the set of ll polynomils with coefficients from the set {, 0, } by F := F n. n=0 More generlly we define the following clsses of polynomils. For M > 0 nd µ 0 let n K µ M := f : fx = j x j, j C, j Mj µ, 0 =, n N. On occsion we let S := S, S M := S M, nd K M := K 0 M. We lso employ the following stndrd nottions. We denote by P n the set of ll polynomils of degree t most n with rel coefficients. We denote by Pn c the set of ll polynomils of degree t most n with complex coefficients. The height of polynomil n p n z := j z j, j C, n 0,

4 4 PETER BORWEIN AND TAMÁS ERDÉLYI is defined by { } j Hp n := mx : j = 0,,..., n. n Also, nd p A := sup pz z A p LqA := pz q dz A /q re used throughout this pper for mesurble functions in this pper usully polynomils p defined on mesurble subset of the unit circle or the rel line, nd for q 0,. 3. New Results The first two results concern lower bounds on subrcs in the supremum norm. Theorem 3.. Let 0 < < 2π nd M. Let A be subrc of the unit circle with length la =. Then there is n bsolute constnt c > 0 such tht c + log M f A exp for every f S M := SM tht is continuous on the closed unit disk nd stisfies fz 0 2 for every z 0 C with z 0 = 4M. Corollry 3.2. Let 0 < < 2π nd M. Let A be subrc of the unit circle with length la =. Then there is n bsolute constnt c > 0 such tht for every f K M := K M. c + log M f A exp The next two results show tht the previous results re, up to constnts, shrp. Theorem 3.3. Let 0 < < 2π. Let A be the subrc of the unit circle with length la =. Then there re bsolute constnts c > 0 nd c 2 > 0 such tht whenever la = c 2. inf f c A exp 0 f F

5 LITTLEWOOD-TYPE PROBLEMS ON SUBARCS OF THE UNIT CIRCLE 5 Theorem 3.4. Let 0 < < 2π nd M. Let A be the subrc of the unit circle with length la =. Then there re bsolute constnts c > 0 nd c 2 > 0 such tht c + log M inf f A exp 0 f K M whenever la = c 2. The next two results extend the first two results to the L norm nd hence to ll L p norms with p. Theorem 3.5. Let 0 < < 2π, M, nd µ =, 2,.... Let A be subrc of the unit circle with length la =. Then there is n bsolute constnt c > 0 such tht c µ + log M f LA exp for every f S µ M tht is continuous on the closed unit disk nd stisfies fz 0 2 for every z 0 C with z 0 4M2 µ. Corollry 3.6. Let 0 < < 2π, M, nd µ =, 2,.... Let A be subrc of the unit circle with length la =. Then there is n bsolute constnt c > 0 such tht c + µ log µ + log M f LA exp for every f K µ M. The following is n interesting consequence of the preceding results. Corollry 3.7. Let A be subrc of the unit circle with length la =. If p k is sequence of monic polynomils tht tends to 0 in L A, then the sequence Hp k of heights tends to. The finl result shows tht the theory does not extend to rbitrry sets of positive mesure. Theorem 3.8. For every ǫ > 0 there is polynomil p K such tht pz < ǫ everywhere on the unit circle except possibly in set of liner mesure t most ǫ. The bove results should be compred with erlier result of the uthors [Bor- 96] on pproximtion on the intervl [0, ]. These stte tht there re bsolute constnts c > 0 nd c 2 > 0 such tht for every n 2. exp c n inf p [0,] exp c 2 n 0 p F n

6 6 PETER BORWEIN AND TAMÁS ERDÉLYI 4. Lemms [ Lemm 4.. Let 0 < < π nd M. Let Γ M be the circle with dimeter + 2M, ]. Let J be the subrc of Γ M with length lj = which is symmetric with respect to the rel line nd contins. Then there is n bsolute constnt c 3 > 0 such tht c3 + log M g J exp for ll g S 4M tht is continuous on the closed unit disk nd stisfies g 4M 4. Our next lemm is known s version of the three-line-theorem. It my be found, for exmple, in [Zy-59, p. 93]. Its proof is so short nd simple tht we present it for the ske of completeness. Lemm 4.2. Let > 0 nd E := {z C : 0 Imz }. Suppose g is n nlytic function in the interior of E, nd suppose g is continuous on E. Then /2 /2 mx gz mx gz mx gz. {z: Imz=/2} {z: Imz=0} {z: Imz=} The next lemm, tht plys crucil role in the proof of Theorem 3., cn be esily derived from Lemm 4.2. Lemm 4.3. Let 0 < < π, α := cos/2+i sin/2, β := cos/2 i sin/2. Let { } α z I t := z C : rg = t. z β Note tht I is the smller rc on the unit circle with endpoints α nd β, nd I 0 is the line segment between α nd β. Suppose g is n nlytic function in the open region bounded by I 0 nd I, nd suppose g is continuous on the closed region between I 0 nd I. Then mx gz z I /2 /2 /2 mx gz mx gz. z I 0 z I To prove Theorem 3.3, we need some corollries of the Hdmrd Three Circles Theorem. Suppose f is regulr inside nd on For r [r, r 2 ], let Then {z C : r z r 2 }. Mr := mx z =r fz. Mr logr2/r Mr logr2/r Mr 2 logr/r.

7 LITTLEWOOD-TYPE PROBLEMS ON SUBARCS OF THE UNIT CIRCLE 7 Corollry 4.4. Let 0, /8]. Suppose f is regulr inside nd on the ellipse E with foci t 8 nd nd with mjor xis [ 4 7, 4 + 7]. Let Ẽ be the ellipse with foci t nd nd with mjor xis Then Corollry 4.5. We hve [ 4 0, 4 + 0]. mx fz mx fz z E e z [ 8,] /2 mx fz n + exp3n /2 z E e for every f F n nd 0, /8]. /2 mx fz. z E mx fz z [ 8,] /2 In one of our proofs of Theorem 3.3 we will need the upper bound of the result below proved in [Bor-96]. An ppliction of this lemm mkes the proof of Theorem 3.3 shorter in the specil cse when the subrc A of the unit circle is symmetric with respect to the rel line nd contins. Our second proof of Theorem 3.3 is longer. This self-contined second proof does not use the lemm below nd elimintes the extr ssumption on the subrc A. Lemm 4.6. There re bsolute constnts c 4 > 0 nd c 5 > 0 such tht exp c 4 n inf p [0,] exp c 5 n 0 p F n for every n 2. To prove Theorem 3.4 we need two lemms. Lemm 4.7. Suppose n px = j x j, j 9, j C, Then n k px = x k qx, qx = b j x j, b j C. n k en k en k q D b j 9n + e 9n + k k where D denotes the unit circle. As consequence, if A denotes the subrc of the unit circle tht is symmetric to the rel line, contins, nd hs length 2k/9n, then e k p A 9n +. 9 To prove Theorem 3.4 our min tool is the next lemm due to Hlász [Tu-84].

8 8 PETER BORWEIN AND TAMÁS ERDÉLYI Lemm 4.8. For every k N, there exists polynomil h P k such tht 2 h0 =, h = 0, hz < exp for z. k Lemm [ 4.9. Let 0 < < π nd M. Let Γ,M,µ be the circle with dimeter + 2M2, cos/4 µ M2 ]. Let I be the subrc of Γ µ,m,µ with length li = which is symmetric with respect to the rel line nd contins cos/4 M2. Then µ there is n bsolute constnt c 4 > 0 such tht c4 µ + log M g I exp for every g S µ M4 µ tht stisfies g 4M2 cos/4 µ 2M2 µ 4. Lemm 4.0. Let w w 2 C nd let z 0 := 2 w + w 2. Assume tht J is n rc tht connects w nd w 2. Let J 2 be the rc tht is the symmetric imge of J with respect to the z 0. Let J := J J 2 be positively oriented. Suppose tht g is n nlytic function inside nd on J. Suppose tht the region inside J contins the disk centered t z 0 with rdius δ > 0. Let gz K for z J 2. Then gz 0 2 πδ K gz dz. J 5. Proofs of Theorems Proof of Lemm 4.. Suppose g S 4M is continuous on the closed unit disk nd g 4M 4. Let 2m 4 be the smllest even integer not less thn 4π/. Let 2πi ξ := exp 2m be the first 2mth root of unity. We define 2m eqully spced points on Γ M by η k := 4M + 4M ξ k, k = 0,,..., 2m. Then there is n bsolute constnt c 5 > 0 such tht z c 5 M k 2, k =, 2,...m, whenever z is on the smller subrc of the circle Γ M with endpoints η k nd η k+ or with endpoints η 2m k nd η 2m k, respectively. We define the function hz := 2m g 4M + 4M ξ j z 4M.

9 LITTLEWOOD-TYPE PROBLEMS ON SUBARCS OF THE UNIT CIRCLE 9 Since g S 4M, we obtin mx z Γ M hz g 2 J m k= 4M c 4 M k c 5 4m 4 4m 4 m Me g 2 πc 5 m c6 + log M exp g 2 J J e 4 em πc 5 4m 4 M 4m 4 m! 4 g 2 J 4m 4 g 2 J with n bsolute constnt c 6 > 0. Now the Mximum Principle yields tht g 2m 4M = h c6 + log M 4M mx hz exp g 2 J. z Γ M Since 2m 2 + 4π/ nd g 4M g 2 c6 + log M J exp c7 + log M exp 4, we obtin g 4M 2m exp c6 + log M with n bsolute constnt c 7 > 0. This finishes the proof. 2m 4 Proof of Lemm 4.2. Let z 0 be complex number with Imz = /2. We wnt to show tht gz 0 2 mx gz mx gz. {z: Imz=0} {z: Imz=} Without loss of generlity we my ssume tht Rez 0 = 0; the generl cse follows by liner trnsformtion. So let z 0 := i/2. Applying the Mximum Principle to hz := gzgi z on E, we obtin gi/2 2 = gi/2gi i/2 = hi/2 mx hz z E = mx z E gzgi z nd the proof is finished. mx gz {z: Imz=0} mx gz {z: Imz=} Proof of Lemm 4.3. This follows from Lemm 4.2 by the substitution α z w := log. z β Proof of Theorem 3.. Without loss of generlity we my ssume tht the rc A is the subrc of the unit circle with length la = < π which is symmetric with respect to the rel line nd contins. Suppose f S M is continuous on the.

10 0 PETER BORWEIN AND TAMÁS ERDÉLYI open unit disk, nd suppose f 4M 2. Using the nottion of Lemm 4.3, let gz := z αz βfz. Then strightforwrd geometric rgument yields tht gz M z αz β z 2M sin/2, z I 0 note tht I 0 is the line segment between α nd β. Hence, with L := g A note tht A = I, we conclude by Lemm 4.3 tht /2 2ML mx gz. z I /2 sin/2 Denote by G the open region bounded by I /2 nd I. By the Mximum Principle mx z G gz mx { L, } /2 2L. sin/2 It is simple elementry geometry to show tht the rc K := Γ M G hs length t lest /2. Here, s in Lemm 4., Γ M denotes the circle with dimeter [ + 2M, ]. Observe tht f S M implies g S 4M. Also, since M, g 4M f 4M Hence Lemm 4. cn be pplied with g S 4M. We conclude tht from which { } /2 2ML c4 + log M mx L, g K exp, sin/2 2 f A 4 g A = L 4 exp c + log M with n bsolute constnt c > 0. Proof of Corollry 3.2. Note tht if f K M, then f S M nd f is continuous on the closed unit disk. Also, if z 0 = 4M, then fz 0 M z 0 z 0 2M 4M = 2. So the ssumptions of Theorem 3. re stisfied nd the corollry follows from Theorem 3.. Proof of Corollry 4.4. This follows from the Hdmrd Three Circles Theorem with the substitution z + z w =

11 LITTLEWOOD-TYPE PROBLEMS ON SUBARCS OF THE UNIT CIRCLE The Hdmrd Three Circles Theorem is pplied with r :=, r := 2, nd r 2 := 4. Proof of Corollry 4.5. This follows from Corollry 4.4 nd the Mximum Principle. We present two proofs of Theorem 3.3. The first one is under the dditionl ssumption tht the subrc A is symmetric with respect to the rel line nd contins. Proof of Theorem 3.3 in the bove specil cse. By Lemm 4.6, for every integer n 2, there is Q n F n such tht Q n [0,] exp c 5 n. Let n be chosen so tht n := N, where N is defined by Then, by Corollry 4.5, = c 5 26 N. mx Q n z n + /2 exp c 5 /2 n /2 mx fz z E e z [ 8,] n + /2 exp c 5 /2 n /2 /2 exp c5 n n + /2 exp c 5 /4 n c6 exp /2 whenever c 7, where c 6 > 0 nd c 7 > 0 re bsolute constnts. Now observe tht the unit circle intersects the ellipse Ẽ in n rc of length t lest c 8, where c 8 > 0 is n bsolute constnt. Therefore the Mximum Principle finishes the proof. Our second proof of Theorem 3.3 is in the generl cse. In ddition, it is selfcontined. Proof of Theorem 3.3. Let A := {e it : t [t, t 2 ]}, where 0 t < t 2 2π nd t t 2 =. Let t 0 := t + t 2 /2, nd w := e it0. We prove the following extension of Lemm 4.6. For every sufficiently lrge n N, there is Q n F n such tht To see this we define k := 2 n. Let be k + equidistnt points. Q n [0,w] exp c 5 n. k/2n =: y 0 < y < < y k :=

12 2 PETER BORWEIN AND TAMÁS ERDÉLYI We use counting rgument to find polynomil f F n with the property 5. fy j w 2 n, j = 0,,..., k, for sufficiently lrge n. Indeed, we cn divide the 2k + dimensionl rel cube Q := {z 0,..., z k C k : Rez j, Imz j [ n, n + } into 2mn + 2k+ subcubes by defining := Q i0,i,...,i 2k+ { [ z 0,..., z k C k i2j : Rez j m, i [ 2j + i2j+, Imz j m m, i } 2j+ +, m where i 0, i,..., i 2k+ re 2k + tuples of integers with n + m i j n + m for ech j = 0,,..., 2k +. Let n A n := f : fx = j x j, j {0, } denote the set of polynomils of degree t most n with coefficients from {0, }. Note tht if P A n, then MP := RePy 0 w, ImPy 0 w,..., RePy k w, ImPy k w Q. Also, there re exctly 2 n elements of A n. Therefore, if 2mn + 2k+ < 2 n holds, then there exist two different P A n nd P 2 A n, nd subcube Q i0,i,...,i 2k+ such tht both nd MP = ReP y 0 w, ImP y 0 w,..., ReP y k w, ImP y k w MP 2 = ReP 2 y 0 w, ImP 2 y 0 w,..., ReP 2 y k w, ImP 2 y k w re in Q i0,i,...,i 2k+, nd hence for 0 f := P P 2 F n, we hve fy j w 2m, j = 0,,..., k. Now choose m := 2 n. This, together with k := 2 n, yields tht the inequlity 2mn 2k+ < 2 n holds provided n is sufficiently lrge. This proves 5.. In the rest of the proof let n be sufficiently lrge to stisfy 5.. Associted with f F n stisfying 5., we define ux := Refxw, x R,

13 LITTLEWOOD-TYPE PROBLEMS ON SUBARCS OF THE UNIT CIRCLE 3 nd Obviously vx := Imfxw, x R. u k+ [0,] n k+2 nd v k+ [0,] n k+2. Let y [y 0, ] be n rbitrry point different from ech y j. By well-known formul for divided differences, uy k k y y j + uy i i=0 y i y k,j i y i y j = k +! uk+ ξ for some ξ [y 0, ]. Combining 5. nd the two observtions bove, we obtin k uy k +! uk+ ξ y y j k +! + i=0 k +! nk+2 + n 2 2n k+ k k uy i y y j y i y k,j i y i y j k k! i!k i! i=0 2 k+ n + 2 n 2 k 2 /2 n n + 2 n 2 /2 n exp c 4 n with n bsolute constnt c 4 > 0. Similrly Hence vy exp c 4 n. fyw 2exp c 4 n with n bsolute constnt c 4 > 0. Since y [y 0, ] is rbitrry, we hve proved tht f [y0w,w] 2 exp c 4 n, where y 0 = k/2n 4 n /2 for sufficiently lrge n. We conclude tht the polynomil gx := x n fx stisfies g F 2n nd g [0,w] exp c 5 n with n bsolute constnt c 5 > 0. Now the proof cn be finished by trivil modifiction of the proof given in the specil cse when subrc A of the unit circle is symmetric with respect to the rel line nd contins.

14 4 PETER BORWEIN AND TAMÁS ERDÉLYI Proof Lemm 4.7. We hve b j = d j j! dx j pxx k x=0 j j k k + m! = p j m 0 j! m k! m=0 j k + m! = k!m! j m! pj m 0 = j k + m! j m k!m! m=0 m=0 j k + m k k + j ek + j = j m 9 9 m k k m=0 en k 9 k which proves the lemm. Proof of Theorem 3.4. Without loss of generlity we my ssume tht A is the subrc of the unit circle with length la = which is symmetric with respect to the rel line nd contins. Let k := Let h Pk be the polynomil with the properties of Lemm 4.8. Let u := k 2. Let Since So Q u x := h k x =: u b j x j. 2 hz < exp, z, k u 4k b j 2 = Q u 2 L exp 2 D = e 4. k b 0 =, b j e 2 9, j = 0,,..., u. Now let A denote the subrc of the unit circle which is symmetric to the rel line, contins, nd hs length 0, 9. Since 2k/9u = 2/9k, Lemm 4.7 implies tht c5 Q u A exp c 4 k exp with some bsolute constnts c 4 > 0 nd c 5 > 0. Now let M. Without loss of generlity we my ssume tht M = 9 m with nonnegtive integer m. When m = 0 the theorem follows from Theorem 3.3. So let m. Let n := um nd let P n x := Q m u x := hkm x =: n j x j. Since hz < exp 2, z, k

15 So LITTLEWOOD-TYPE PROBLEMS ON SUBARCS OF THE UNIT CIRCLE 5 n Also, using m, we obtin 4km j 2 = P n 2 L exp 2 D = e 4m. k 0 =, j e 2m M, j = 0,,..., n. P n A = Q u m A = exp c5 m c2 + log M exp with n bsolute constnt c 2 > 0. This finishes the proof. Proof of Lemm 4.9. The proof is the sme s tht of Lemm 4. with trivil modifictions. Proof of Lemm 4.0. Applying Cuchy s integrl formul with on J, we obtin Gz := gz 0 + z z 0 gz 0 z 0 z gz 0 2 = Gz 0 = Gz dz 2πi J z z 0 = 2 Gzdz 2π J z z 0 Gz dz π J z z 0 = gz 0 + z z 0 gz 0 z z 0 dz π J z z 0 πδ K gz dz. J Proof of Theorem 3.5. Without loss of generlity we my ssume tht the rc A is the subrc of the unit circle with length la = < π/2 which is symmetric with respect to the rel line nd contins. Suppose f S µ M nd f 4M2 cos/4 µ 2M2 µ Note tht this is gurnteed by the ssumption of the theorem since Let the region H,M,µ be defined by H,M,µ := M2 µ cos/4 2M2 µ 4M2 µ, { z = re iθ : cos/4 < r < cos/4 M2 µ, 4 < θ < }. 4

16 6 PETER BORWEIN AND TAMÁS ERDÉLYI Let Γ,M,µ be the circle s in Lemm 4.9. It is simple geometric rgument to show tht the rc I := Γ,M,µ H,M,µ hs length greter thn c 5 with n bsolute constnt c 5 > 0. Let z 0 I H,M,µ be fixed. Then we cn choose w A nd w 2 A such tht z 0 = 2 w + w 2. Let J be the rc connecting w nd w 2 on the unit circle. Note tht J is subrc of A. Let J 2 be the rc which is the symmetric imge of J with respect to the line segment connecting w nd w 2. Let gz := z w z w 2 µ fz. Then it is elementry geometry gin to show tht gz M z w z w 2 µ z µ M2µ sin µ /2, z J 2. By Lemm 4.0 we obtin π cos/4 5.2 gz 0 2 M2 µ M2 µ sin µ gz dz. /2 J Observe tht f S µ M implies g Sµ M4 µ. Also, since M, g 4M2 cos/4 µ 2M2 2 µ 2 2µ f µ 2µ f 8µ M2 cos/4 µ 2M2 µ 4M2 cos/4 µ 2M2 µ Hence Lemm 4.9 cn be pplied with g S µ M4 µ. We conclude tht there is point z 0 I H,M,µ such tht c4 µ + log M gz 0 exp. Combining this with 5.2 nd J A gives µ µ f LA g LA g LJ 4 4 µ π cos/4 sin µ /2 4 M2 µ M2 µ gz 0 2 c µ + log M exp with n bsolute constnt c > 0. Proof of Corollry 3.6. Let f K µ M. Then f Sµ Mµ! 2 nd f is continuous on the closed unit disk. Also, if z 0 4Mµ! 2 2 µ, then j fz 0 Mµ! 2 j µ 4Mµ! 2 2 µ Mµ!2 4Mµ! 2 4 j= j 2 j j= j= µ j 2 j

17 LITTLEWOOD-TYPE PROBLEMS ON SUBARCS OF THE UNIT CIRCLE 7 So the ssumptions of Theorem 3.5 re stisfied with M replced by Mµ! 2, nd the corollry follows from Theorem 3.5. Proof of Corollry 3.7. Let n k p k z = j,k z j, j,k C, nk,k 0, nd let M k := Hp k. Applying Corollry 3.6 with q k z := nk,k z n k p k z K Mk = K M k nd the rc B := {z : z A} of length, we obtin the corollry. 6. Proof of Theorem 3.8 Lemm 6.. For every r 0, /2 there exists trigonometric polynomil pz = n j= n such tht c 0 =, c j < r nd pz < r everywhere on the unit circle except possibly in set of liner mesure t most r. Proof. The finite Riesz product pz = c j z j N + rz mj + rz mj j= with m j := 4 j nd sufficiently lrge N is such n exmple. For r 0, /2 nd m j = 4 j the Riesz products tend to 0 lmost everywhere on the unit circle s N. See, for exmple, [Zy-59, p 208]. The next lemm follows simply from the fct tht the trnsfinite dimeter of ny closed proper subset of the unit circle is less thn. We remrk tht due to this fct the polynomil gurnteed by Lemm 6.2 cn be chosen so tht its coefficients re integers. We will not need this extr property. Lemm 6.2. For every η > 0 there exists polynomil gz = L b k z k k=0 such tht b 0 = nd gz < η everywhere on the unit circle except possibly on set of liner mesure t most η.

18 8 PETER BORWEIN AND TAMÁS ERDÉLYI Proof of Theorem 3.8. For η := ǫ/2 we choose polynomil L gz = b k z k k=0 with the properties of Lemm 6.2, tht is, b 0 = nd 6. gz < ǫ 2 everywhere on the unit circle except possibly in set of liner mesure t most ǫ 2. For every k with b k > we choose trigonometric polynomil so tht nd 6.2 p k z < ǫ 2L b k p k z = n k j= n k c j,k z j c 0,k =, c j,k < ǫ 2L b k everywhere on the unit circle except possibly in set of liner mesure t most This cn be done by Lemm 6.2. Now let hz := g z A L = Finlly we define k=0 ǫ 2L. b k z Ak with A := + 2 mx k: b k > n k. fz := hz L k= b k > b k z Ak p k z. It is strightforwrd from the construction tht f K. Also, 6. nd the definition of h imply tht 6. hz < ǫ 2 everywhere on the unit circle except possibly in set of liner mesure t most ǫ 2. Finlly 6.2 nd the definition of f imply tht fz < ǫ 2 + L k= b k > ǫ b k 2L b k ǫ 2 + L ǫ 2L = ǫ everywhere on the unit circle except possibly in set of liner mesure t most ǫ 2 + L ǫ 2L = ǫ. This finishes the proof. 7. Acknowledgment. We thnk Fedor Nzrov for observing nd proving Theorem 3.8. The content of Section 6 is due to him. We lso thnk him for severl discussions bout problems relted to the pper.

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