Scholarlink

<p>Let <i>Z</i>(<i>N</i>) denote the minimum number of zeros in [0, 2<i>π</i>] that a cosine polynomial of the form <Equation ID="Equ1"> <EquationSource Format="TEX">\(f_{A}(t)={\sum_{n \in A}} \cos nt\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mi>f</mi> <mrow> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <munder> <mo>∑</mo> <mrow> <mi>n</mi> <mo>∈</mo> <mi>A</mi> </mrow> </munder> </mrow> <mi>cos</mi> <mspace width="thinmathspace" /> <mi>n</mi> <mi>t</mi> </math></EquationSource> </Equation> can have when <i>A</i> is a finite set of non-negative integers of size ∣<i>A</i>∣ = <i>N</i>. It is an old problem of Littlewood to determine <i>Z</i>(<i>N</i>). In this paper, we obtain the lower bound <i>Z</i>(<i>N</i>) ≽ (log log <i>N</i>)<sup>(1+<i>o</i>(1))</sup> which exponentially improves on the previous best bounds of the form <i>Z</i>(<i>N</i>) ≽ (log log log <i>N</i>)<sup><i>c</i></sup> due to Erdélyi and Sahasrabudhe.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

An improved lower bound for a problem of Littlewood on the zeros of cosine polynomials

  • 期刊论文

  • 出版模式：

    开放获取

  • 发表日期：

    2025年11月30日

查看全文
  • Benjamin Bedert

摘要

Let Z(N) denote the minimum number of zeros in [0, 2π] that a cosine polynomial of the form \(f_{A}(t)={\sum_{n \in A}} \cos nt\) f A ( t ) = ∑ n ∈ A cos n t can have when A is a finite set of non-negative integers of size ∣A∣ = N. It is an old problem of Littlewood to determine Z(N). In this paper, we obtain the lower bound Z(N) ≽ (log log N)(1+o(1)) which exponentially improves on the previous best bounds of the form Z(N) ≽ (log log log N)c due to Erdélyi and Sahasrabudhe.