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X-minimal patterns and a generalization of Sharkovskiĭ's theorem

Jozef Bobok, Milan Kuchta (1998)

Fundamenta Mathematicae

We study the law of coexistence of different types of cycles for a continuous map of the interval. For this we introduce the notion of eccentricity of a pattern and characterize those patterns with a given eccentricity that are simplest from the point of view of the forcing relation. We call these patterns X-minimal. We obtain a generalization of Sharkovskiĭ's Theorem where the notion of period is replaced by the notion of eccentricity.

Zeros of Fekete polynomials

Brian Conrey, Andrew Granville, Bjorn Poonen, K. Soundararajan (2000)

Annales de l'institut Fourier

For p an odd prime, we show that the Fekete polynomial f p ( t ) = ∑ a = 0 p - 1 a p t a has ∼ κ 0 p zeros on the unit circle, where 0 . 500813 > κ 0 > 0 . 500668 . Here κ 0 - 1 / 2 is the probability that the function 1 / x + 1 / ( 1 - x ) + ∑ n ∈ ℤ : n ≠ 0 , 1 δ n / ( x - n ) has a zero in ] 0 , 1 [ , where each δ n is ± 1 with y 1 / 2 . In fact f p ( t ) has absolute value p at each primitive p th root of unity, and we show that if | f p ( e ( 2 i π ( K + τ ) / p ) ) | < ϵ p for some τ ∈ ] 0 , 1 [ then there is a zero of f close to this arc.

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