Counting zeros of cosine polynomials: On a problem of Littlewood

We show that if A is a finite set of non-negative integers then the number of zeros of the functionfA(θ)=∑a∈Acos⁡(aθ), in [0,2π], is at least (log⁡log⁡log⁡|A|)1/2−ε. This gives the first unconditional lower bound on a problem of Littlewood and solves a conjecture of Borwein, Erdélyi, Ferguson and Lo...

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Bibliographic Details
Published in:Advances in mathematics (New York. 1965) 2019-02, Vol.343, p.495-521
Main Author: Sahasrabudhe, Julian
Format: Article
Language:English
Subjects:
Exponential sums
Fourier series
Littlewood
Polynomials
Reciprocal polynomials
Trigonometric polynomials
Zeros of polynomials
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Summary:We show that if A is a finite set of non-negative integers then the number of zeros of the functionfA(θ)=∑a∈Acos⁡(aθ), in [0,2π], is at least (log⁡log⁡log⁡|A|)1/2−ε. This gives the first unconditional lower bound on a problem of Littlewood and solves a conjecture of Borwein, Erdélyi, Ferguson and Lockhart. We also prove a result that applies to more general cosine polynomials with few distinct rational coefficients. One of the main ingredients in the proof is perhaps of independent interest: we show that if f is an exponential polynomial with few distinct integer coefficients and f “correlates” with a low-degree exponential polynomial P, then f has a very particular structure. This result allows us to prove a structure theorem for trigonometric polynomials with few zeros in [0,2π] and restricted coefficients.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2018.11.025