On Polynomials with Coefficients of Modulus One

The norm of the Fourier transform on finite abelian groups

“…This result was extended by Byrnes to a wider class of biunimodular functions (see [9]). Littlewood was interested in proving (7.4) for Gauss sequences.…”

On Bernstein's Inequality and Kahane's Ultraflat Polynomials

“…Indeed, his proof was inspired by the (probabilistic and nonprobabilistic) arguments of Körner's paper [18], in which the main object was to prove another conjecture of Littlewood somewhat weaker than the existence of ultraflat polynomials in G n . The method of proof (and chief merit) of Körner's paper [18] consisted of adding probabilistic ideas to an ingenious (nonprobabilistic) construction of Byrnes [8,Theorem 2]. However, in the spring of 1995, that is, nearly two decades after the publication of [8], John J. Benedetto [5] and his student Hui-Chuan Wu made an astonishing discovery: Theorem 2 of [8], which Körner used in [18], was erroneous.…”

“…The method of proof (and chief merit) of Körner's paper [18] consisted of adding probabilistic ideas to an ingenious (nonprobabilistic) construction of Byrnes [8,Theorem 2]. However, in the spring of 1995, that is, nearly two decades after the publication of [8], John J. Benedetto [5] and his student Hui-Chuan Wu made an astonishing discovery: Theorem 2 of [8], which Körner used in [18], was erroneous. (Based on extensive numerical calculations, George Benke [6] had suspected such an error as early as 1987.)…”

“…The nonprobabilistic arguments in Kahane [16], although indirectly inspired by Körner [18], are based on other analytic tools unrelated to [18]. In particular, they make no use of the mistaken Theorem 2 of [8].…”

Asymptotic Evaluation of Gauss Polynomials

“…Although the (b(a) do not satisfy (2), Littlewood showed that if the sum in <j>(a) is taken to«-1, these polynomials do satisfy (2) in intervals n~x/2+s <| a |*£ {-for every 8 > 0. Byrnes [1] has given other polynomials with coefficients of modulus 1 also failing the uniformity criterion in a similar way. More recently, J.-P. Kahane has given a probabilistic proof of the existence of such polynomials satisfying (2).…”