Lower bounds for the number of zeros of cosine polynomials in the period: a problem of Littlewood

Borwein, Peter; Erdélyi, Tamás

doi:10.4064/aa128-4-5

Cited by 16 publications

(18 citation statements)

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“…This follows as a consequence of a more general result, see Corollary 2.3 in [34], stated as Corollary 1.5 here, in which the coefficients of the self-reciprocal polynomials P n of degree at most n belong to a fixed finite set of real numbers. In [7] we proved the following result. Theorem 1.1.…”

Section: Introduction and Notationmentioning

confidence: 77%

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Improved lower bound for the number of unimodular zeros of self-reciprocal polynomials with coefficients in a finite set

Erdélyi

2019

Acta Arith.

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Let n 1 < n 2 < · · · < n N be non-negative integers. In a private communication Brian Conrey asked how fast the number of real zeros of the trigonometric polynomials T N (θ) = N j=1 cos(n j θ) tends to ∞ as a function of N . Conrey's question in general does not appear to be easy. Let P n (S) be the set of all algebraic polynomials of degree at most n with each of their coefficients in S. For a finite set S ⊂ C let M = M (S) := max{|z| : z ∈ S}. It has been shown recently that if S ⊂ R is a finite set and (P n ) is a sequence of self-reciprocal polynomials P n ∈ P n (S) with |P n (1)| tending to ∞, then the number of zeros of P n on the unit circle also tends to ∞. In this paper we show that if S ⊂ Z is a finite set, then every self-reciprocal polynomial P ∈ P n (S) has at least c(log log log |P (1)|) 1−ε − 1 zeros on the unit circle of C with a constant c > 0 depending only on ε > 0 and M = M (S). Our new result improves the exponent 1/2 − ε in a recent result by Julian Sahasrabudhe to 1 − ε. Sahasrabudhe's new idea [66] is combined with the approach used in [34] offering an essencially simplified way to achieve our improvement. We note that in both Sahasrabudhe's paper and our paper the assumption that the finite set S contains only integers is deeply exploited.

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Section: Introduction and Notationmentioning

confidence: 77%

“…In the papers [7], [34], and [66] the already mentioned Littlewood Conjecture, proved by Konyagin [45] and independently by McGehee, Pigno, and Smith [55], plays a key role, and we rely on it heavily in the proof of the main results of this paper as well. This states the following.…”

Section: Introduction and Notationmentioning

confidence: 99%

Improved lower bound for the number of unimodular zeros of self-reciprocal polynomials with coefficients in a finite set

Erdélyi

2019

Acta Arith.

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“…Progress on Littlewood's question of lower bounds was achieved by Borwein and Erdélyi [7] who showed that if A ⊆ N ∪ {0} is an infinite set and A n = A ∩ [1, n], for each n ∈ N, then the number of zeros of f An tends to infinity with n. Interestingly, their argument is both ineffective and depends crucially on the nested structure of the sets A n and thus does not imply that the roots of a general sequence of polynomials tends to infinity. In this paper we answer the question of Borwein, Erdélyi, Ferguson and Lockhart in a strong form: by giving an explicit lower bound on the number of zeros for Littlewood's problem.…”

Section: Resultsmentioning

confidence: 99%

Counting zeros of cosine polynomials: On a problem of Littlewood

Sahasrabudhe

2019

Advances in Mathematics

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We show that if A is a finite set of non-negative integers then the number of zeros of the function f A (θ) = a∈A cos(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 coefficents.is well known that any continuous, even function f : R/(2πZ) → R can be uniformly approximated by cosine polynomials. Thus, for large N , the behavior of such polynomials is essentially as rich as the class of continuous functions. In what follows, we consider what can be said about the behavior of the function f if the coefficients are restricted to a small set.One of the early champions of questions about polynomials with restricted coefficients was J.E.Littlewood, who studied many problems of this general type and stimulated the field with a wealth of problems [23,24,25,26]. Perhaps most famously, Littlewood asked for the minimum L 1 -norm attained by a function of the form a∈A cos(aθ), where A ⊆ N ∪ {0} is of a given size. This question was resolved (up to constants) in 1981 independently by Konyagin [22], and McGehee, Pigno, and Smith [28] after a series of partial results were obtained by Salem, Cohen, Davenport and Pichorides [33,8,9,32]. We note that these estimates have seen numerous applications and modifications in recent years.Counting zeros of polynomials with restricted coefficients has also received considerable attention;after the seminal work of Littlewood and Offord [27], several results have been proved on the number of roots of "typical" polynomials with restricted coefficients. For example, Kac [20,21] has given an exact integral formula for the expected number of roots of a degree N polynomial with coefficients sampled identically and independently from a Gaussian distribution. In a similar vein, Erdős and Offord [15] showed that almost all polynomials of the form N i=0 ε i x i , where ε 0 , . . . , ε N ∈ {−1, +1}, have ( 2 π + o(1)) log N real roots. These results have been subsequently refined and extended by various authors. For more, we refer the reader to the recent paper of Hoi Nguyen, Oanh Nguyen and Van Vu [30].The study of extremal questions on the number of real zeros of polynomials with restricted coefficients began with the work of Bloch and Pólya [2] in 1932, who studied the maximum number of real zeros attainable by polynomials of the form N i=0 ε i x i , ε 0 , . . . , ε N ∈ {−1, 0, +1}. After improvements by other authors [34,35], Erdős and Turán proved a landmark ...

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“…This does not appear to be easy. The case when the sequence 0 ≤ n 1 < n 2 < · · · is fixed was handled in [3].…”

Section: Resultsmentioning

confidence: 99%

On the zeros of cosine polynomials: solution to a problem of Littlewood

et al. 2008

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Lower bounds for the number of zeros of cosine polynomials in the period: a problem of Littlewood

Cited by 16 publications

References 11 publications

Improved lower bound for the number of unimodular zeros of self-reciprocal polynomials with coefficients in a finite set

Improved lower bound for the number of unimodular zeros of self-reciprocal polynomials with coefficients in a finite set

Counting zeros of cosine polynomials: On a problem of Littlewood

On the zeros of cosine polynomials: solution to a problem of Littlewood

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