Polynomial Norm Problems in Number Theory

在这个方案中，我们研究了具有限制系数的多项式。特别地，我们感兴趣的是以下几类多项式：单模多项式，Littlewood多项式和Newman多项式，它们的系数分别是单位圆上的复数，来自{-1，1}和来自{0，1}。在这个项目中，我们感兴趣地研究了各种非负p的这些多项式中单位圆上的Lp范数的分布。在多项式的研究中，最重要的结果之一是Kahane关于超平坦么模多项式的定理。Kahane定理指出，存在单模多项式序列{Pn}和实数序列{En}，使得(1)：1-En(Z)|/n1/2n对所有|z|=1，且En趋于0。人们很自然地会问，卡恩定理是否也适用于利特尔伍德多项式。Littlewood提出了一个猜想：(1)对Littlewood多项式序列成立，而Erdos猜想则相反。Peter Borwein和提出者一直在深入研究Littlewood多项式的L4范数，鉴于Erdos的猜想，我们提出猜想(L4范数猜想)：对于任何Littlewood多项式序列{Pn}，存在常数c&gt；0使得我们有(1c)n1/2 4(Pn)。L4范数猜想显然暗示了鄂尔多斯的猜想。此外，我们的猜想还暗示长度为偶数的Barker序列是不存在的。最近，Erdelyi和提出者证明了Littlewood多项式的L4范数的极限分布是一点分布。这一结果给出了大量的Littlewood多项式，其L4范数接近其平均值。我们计划将这一结果推广到Littlewood多项式的其他子类，并希望找到更小的L4范数的平均值。这将揭示L4范数猜想。多项式P的*Mahler测度可被认为是L0(P)。关于Mahler测度最著名的问题是Lehmer问题，该问题询问是否存在c&gt；0使得(2)：1c0(P)。莱默的问题仍然悬而未决。作为L4范数的极限分布，Erdelyi和作者还证明了Littlewood多项式的Mahler测度的极限分布是单点分布。这一结果使我们获得了Littlewood多项式具有小Mahler测度的新记录。我们建议修正我们的论点，以发现更多具有更小Mahler测度的特殊Littlewood多项式。在Erdelyi和将发表在Matheatische Zeitschrift上的一篇最近的论文中，我们研究了Newman多项式的Lp范数，这与Bourain的一个猜想有关。我们计划进一步研究Newman多项式的Lp范数的极限分布，并有望解决Bourain的猜想。

In this proposal, we study polynomials with restricted coefficients. In particular, we are interested in  the following classes of polynomials: Unimodular Polynomials, Littlewood Polynomials and Newman Polynomials, which are the polynomials whose coefficients are respectively complex numbers on unit circle, from {-1,+1} and from {0, +1}. In this project, we are interested in studying the distributions of Lp norms over the unit circle among these classes of polynomials for various non-negative p.*******In the study of polynomials, one of the most important results is Kahane's theorem on the ultra-flat unimodular polynomials. Kahane's theorem states that there are sequences of unimodular polynomials {Pn} and of real numbers {En} such that (1): 1-En n(z)|/n1/2n  for all |z|=1 and En tends to 0. It is natural to ask if Kahane's Theorem is also true for Littlewood polynomials. Littlewood made a conjecture that (1) holds for sequences of Littlewood polynomials, while Erdos conjectured the converse. Either of these two is still wide open.*******Peter Borwein and the proposer have been studying the L4 norm of Littlewood polynomials intensively and in view of Erdos' conjecture, we made the conjecture (L4 Norm Conjecture) that there is a constant c>0 such that for any sequence of Littlewood polynomials {Pn} we have (1+c)n1/2 4(Pn). The L4 Norm Conjecture clearly implies Erdos' conjecture. Moreover, our conjecture also implies the non-existence of Barker sequences with even length. This conjecture suggests finding Littlewood polynomials with small Lp norm.*******Very recently, Erdelyi and the proposer showed that the limiting distribution for the L4 norm of Littlewood polynomials is the one-point distribution. This result gives us plenty of Littlewood polynomials with the L4 norm close to its mean value. We plan to extend this result to other subclasses of Littlewood polynomials and hope to find smaller mean values of the L4 norm.  This will shed light on the L4 norm conjecture.*** ***Mahler measure of a polynomial P can be regarded as L0(P). The most famous problem with Mahler measure is Lehmer's problem which asks if there is c>0 such that (2): 1+c 0(P). Lehmer's problem is still wide open. As the limiting distribution for the L4 norm, Erdelyi and the proposer also show that the limiting distribution for Mahler measure of Littlewood polynomials is the one-point distribution. This result enables us to obtain a new record of having small Mahler measures for Littlewood polynomials. We propose to modify our argument to discover more special Littlewood polynomials with smaller Mahler measures.*** ***In a recent paper of Erdelyi and the proposer which will appear in Mathematische Zeitschrift, we study the Lp norm of the Newman Polynomials which relates to a conjecture of Bourgain. We plan to study further the limiting distribution of the Lp norm of the Newman Polynomials and hopefully settle Bourgain's conjecture. *****