Cosine polynomials with few zeros
| Autor: | Tomas Juškevičius, Julian Sahasrabudhe |
|---|---|
| Přispěvatelé: | Apollo - University of Cambridge Repository |
| Rok vydání: | 2021 |
| Předmět: |
cosine polynomials
old conjecture of Littlewood analysis of their constructions General Mathematics Mathematics::Classical Analysis and ODEs 11C08 41A17 26C10 30C15 05D99 60B15 60C05 01 natural sciences Combinatorics 30C15 (primary) 4903 Numerical and Computational Mathematics Classical Analysis and ODEs (math.CA) FOS: Mathematics Mathematics - Combinatorics Trigonometric functions 60C05 (secondary) Number Theory (math.NT) 0101 mathematics Research Articles Mathematics Conjecture Mathematics - Number Theory 4901 Applied Mathematics 010102 general mathematics 4904 Pure Mathematics Binary logarithm Mathematics - Classical Analysis and ODEs 11C08 49 Mathematical Sciences Combinatorics (math.CO) 26C10 60B15 Research Article |
| Zdroj: | Bulletin of the London Mathematical Society, Hoboken : Wiley, 2021, vol. 53, iss. 3, p. 877-892 |
| ISSN: | 1469-2120 0024-6093 |
| DOI: | 10.1112/blms.12468 |
| Popis: | In a celebrated paper, Borwein, Erd\'elyi, Ferguson and Lockhart constructed cosine polynomials of the form \[ f_A(x) = \sum_{a \in A} \cos(ax), \] with $A\subseteq \mathbb{N}$, $|A|= n$ and as few as $n^{5/6+o(1)}$ zeros in $[0,2\pi]$, thereby disproving an old conjecture of J.E. Littlewood. Here we give a sharp analysis of their constructions and, as a result, prove that there exist examples with as few as $C(n\log n)^{2/3}$ roots. Comment: 17 pages. A few typos fixed |
| Databáze: | OpenAIRE |
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