Byrnes, J. S.

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ON POLYNOMIALS WITH COEFFICIENTS OFMODULUS ONEJ. S. BYRNESIntroductionAs in Littlewood [5], we let ^ ,, be the class of all polynomials of the formngn(Q)= I exp(afci)z\fc = Owhere the afe are arbitrary real constants and z = exp(27i/0). Clearly \\gn\\Lt = (n +1)1for all gne&n, and the question "how close can such a gn come to satisfying|g,,| = (n+1)*?" has long been the object of intense study. In [5] Littlewoodconjectured that there are positive absolute constants Ak and A2 such that, forarbitrarily large n, there exist gnt&n with ^ ^ ^ \gn(0)\ ^ A2n* for all 0. In [3]Erdos conjectured that there is a universal constant c > 0 such that for n ^  2,* for all &,€*„.It was shown by Littlewood [5] that the functiong(6)= £m = 0satisfies: (i) for any d > 0, \g\n~* -» 1 uniformly in n~i+d ^ \6\ ^ -^ , and(») l#l ^ l#4n* for all 0. In the first part of this paper we strengthen part (i) of theLittlewood result in two ways by producing polynomials g which yield an improvedestimate for |g|n~* in a larger subset of the unit circle. In the second part we use themethods already developed to construct functions which are " almost" in ^,, andwhich satisfy the Littlewood conjecture with (within the error) Ax = A2 = 1.1. To begin our work, we require two elementary lemmas.LEMMA 1. LetF(x,T)=(l-e2«ix) 2m=o m+ Twhere T > 1 and F is defined by continuity when x is an integer. Then \F(x, T)\ < 3/7for all x.Proof of Lemma 1. Let x be fixed and not an integer. ThenR e2nimx j 1 °° g2n imxF(x, T) = lim (i-e2nix) 27 m = i (m2-™)7 + 2 m + 7 - lReceived 4 September, .1975; revised 20 May, 1976.[BULL. LONDON MATH. SOC, 9 (1977), 171-176]172 J. S. BYRNESand „, i rTiJ xzXTcompleting the proof of Lemma 1.LEMMA 2. / / r and x are not integers, then« e2nimx 2nie2nikr£ = 1^'m = -oo m-\-r \—ewhere k = \x\.of Lemma 2. Simply compute the Fourier Series of the function F(x) ofperiod 1 which, for 0 < x < 1, is given by2ni . .\-e-2n,rEmploying these Lemmas, we are able to prove our basic result.THEOREM 1. Let N be a positive integer, and define the functionN-lN-lPe^Ni-iby P(9)= £ £ exp(2nijkN-i)zj+kN, z = exp(27H0).Then(a) Pi-^r = N for all integers j ,(b)For any e, N~1<e<$, | P ( 0 ) | = J V + £ for - l + e^O ^ -e, where\E\ < 1 1/ 1 \ /N-l\(c) For N odd, P\— = 0(1), while for N even, P ) = 0(1), and\2N J \ 2N2 )(d) \P(0)\ < 12+ —\ N+ 0(1) for all 9.Proof of Theorem 1. A straightforward calculation shows that for integersm, r with 0 ^ m, r < N we haveand (a) follows.To obtain (b) we define the functionsiN2n * = -oo k+N6and. • \T v—i ~kNiNH(0) = — (1 -zNKJ 2(  ) £ ,n fc = o k+NOON POLYNOMIALS WITH COEFFICIENTS OF MODULUS ONEand the error functionsR(9) = P(6)-H(0) and S(0) = H(0)-G(0).To estimate R(6) we let173sin x 1a n d1 i 12(1 - cos x) xWe can now write1— exp(/x) x — + '</> (*).R(9) = (l-zN[k=0which, together with the facts that <f)'(x) < 0, — 2n < x < 2n, and — 1 +e ^ 0 ^ — E,E < % yieldsN-lsk=0(*?+•"->N - lin (^ -4> (in— +e))l1 / 1 1 \V 2TT \ l+0 0/127Tfi(l-e)2 2—+ —It TIE(1)To estimate S(0), we observe that 1 -NO ^ EN > 1 and A (^l +0) ^ eiV > 1, sothat we may apply Lemma 1 with x = ±N9 and T = EN to getfcN- i zy= -oo A:+iV0TkJV+ I,2«Wfl| II y2TT " ' l|fe=0 A:+1-N0 = ok+N(l+9) j (2)Tie(3)Finally, combining (1), (2) and (3) with the fact that P(9) = G(9) + R(9) + S(9),we obtain (b).Next we apply Lemma 2 with x = r = NO to conclude that=Nforal l0.174 J. S. BYRNESFor (c) we assume that |0| < \ and apply the techniques of the proof of (b) to get/JV /[±JV] e2"ikN0 N-l elni\N9 vP(0) = — (1 -e2«m) I + £ +0(1)2n U = o k+NO *=[iJv]+i k-N+NOJ:\T . oo 2nikN0 _t 2nikN0x= — (1 -e2"1"0) 2 +*2'""20 I +0(1) (4)If N is odd, (4) immediately implies thatP (—) = 0(1).\2NJFor N even, (4) yields/ N - U 2iN_ / « {•P l 2 N 2 ) = * l ? o 2Ar-hl fc=o 2A:+12/JV .,„ « ( - l ) k cos (kn/N) 4N » ( - l ) k s i n (A:7r/iV)_ — n —en>/) y -— —'—- + — y ——-—-—•—-n \=o 2k+\ n k = i 2k+l+ 0(1) = 0(1),where the final estimate follows, for example, from Gradshteyn and Ryzhik [4], p. 38,formulas 1.442-3 and 4, and (c) is proven.To establish (d) we may assume that |0| < £ , and we let NO = M + t, whereM = [NO]. For M ^ - 1 we have, by Lemmas 1 and 2,(i-e2niN0)00 £ 2nikN0fc = o k+NO(12nikt Af -1( oo e2nikt  e2iri/ct v II " £ " <2TT + 3. (5)jt = -oo k + t fc = -oo k + t }\It is equally trivial to obtain an identical estimate for M ^ 0, and the same methodalso yields<2n + 3 for |0| < ±. (6)Finally, (d) follows from (4), (5), and (6), and the proof of Theorem 1 is complete.We point out several immediate consequences of this theorem. First, if a is a fixedreal number and if we define Q by Q(0) = P(0 + a), it is obvious that Q e ^ . ] .Therefore, the bad interval in (b) can be shifted to any interval of length 2e.Second, if we are interested in a fixed subinterval of the unit circle, the estimate in(b) becomes quite remarkable. For example, setting e = i we obtainCOROLLARY 1. On the unit semicircle — | ^ 0 ^ —i we have \P(0)\ =N + E, where\E\ < 1+227T1 < 9 .Third, by employing, for example, a result of Beller [1], we are able to extendTheorem 1 to the case of polynomials of arbitrary degree. We haveON POLYNOMIALS WITH COEFFICIENTS OF MODULUS ONE 175COROLLARY 2. Let n be a positive integer. Then there is a g&^n satisfying(e) For any e, [n*]"1 < £ < \, |g(0)| = n* + E for - 1 +£ < 0 ^ - £ ,( 0 l*(0)l < (2 + 3/7r)w* + 2w± + O(l) for all 0.Proof of Corollary 2. LetiV = [w*], /w = n-N2, and choose P(0)as in Theorem 1.By Beller's result [1], we can choose/e^ m such that |/(0)| < W72m* < 2AI* for all 0.If we now let g(6) = P(6)+zN2f(0), the required estimates follow immediately fromthe Theorem, and Corollary 2 is proven.Finally we observe that if we choose e in Corollary 2 to be, for example, n~* log n,we obtain the improvement of Littlewood's result mentioned in the introduction.2. We now proceed with our construction of functions G(0) which are almost in&„, and which satisfy |G(0)| = w* + O(/i*) for all 0. Toward this end, we haveTHEOREM 2. Let n be a positive integer, and let N be the even positive integersatisfying N2 ^ n < (N + 2)2. Then there exist functions / and g such that(A) z^f+geV,,, and(B) |/(0)+g(0)| = «* + O(w*), where the error is uniform in 6 and n.Remark. It will be seen from the following construction that g consists of twoparts; a polynomial in 0/I_2,I3/4 + O(ni/2), plus z to an integral power multiplied by apolynomial with coefficients of modulus \ and degree 4«* + O(/?*) in z*. Also, / is afunction of precisely the same type as the second part of g, just described. Thus wesee that, except for a relatively small number (i.e., O(n*)) of terms, f+g£&n, and sowe can u s e / + g as the function G(0) mentioned above.Proof of Theorem 2. Define 8 by SN = [JV±], let z = exp(27r/0), - £ < 0 < £, letm = n—N2, and choose Gx&^m such that |G1(0)| = Oty*) (see the proof ofCorollary 2). Define the functions2N-1zjN £fc=Oi 2N-1F(0)=z^N-dN)N h £ zJN £I k=0N-l \zJN £ exp2Jbr/(/Ar1+0)k = 0 }and176 ON POLYNOMIALS WITH COEFFICIENTS OF MODULUS ONEA straightforward calculation yields (A). To establish (B) we proceed as in theproof of Theorem 1 (b), and we obtain/ iN+SN-l ZjN-zN)— i —In j = -iN-5zJN2n j = -<»j +0{n*).Therefore, by Lemma 2, \f+g\ = JV + 0(w*) = n* + O(n% and the proof ofTheorem 2 is complete.In conclusion we mention a result with a similar flavour to Theorem 2 by Bellerand Newman [2], who prove the Littlewood conjecture, with AL and A2 both muchsmaller than 1, for polynomials whose coefficients are bounded by 1 in modulus.References1. E. Beller, " Polynomial Extremal Problems in L", " Proc. Amer. Math. Soc, 30 (1971), 247-259.2. and D. J. Newman, " The Minimum Modulus of Polynomials ", Proc. Amer. Math. Soc,45 (1974), 463-465.3. Paul Erdos, " Some Unsolved Problems ", Michigan Math. J., 4 (1957), 291-300.4. I. S. Gradshteyn, and I. M. Ryzhik, Table of Integrals, Series, and Products (New York, AcademicPress, 1965).5. J. E. Littlewood, " On Polynomials J. ± zm, J, e"-' zm, z = e'°", J. London Math. Soc, 41(1966), 367-376. ° °6. G. G. Lorentz, Approximation of Functions (New York, Holt, Rinehart and Winston, 1966).Forschungsinstitut Fiir Mathematik,ETH,ZurichandUniversity of Massachusetts,Boston

On Polynomials with Coefficients of Modulus One

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