A positive lower bound for \liminf_N→∞ ∏_r=1^N | 2sin πr φ|

2018/10/04 by Sigrid Grepstad, Grepstad, Sigrid, Lisa Kaltenböck +3
Physics and Astronomy · Mathematics · #Advanced Mathematical Theories and Applications #Analytic Number Theory Research #History and Theory of Mathematics

paper · pdf · doi:10.48550/arxiv.1810.02301

Abstract

Nearly 60 years ago, Erdős and Szekeres raised the question of whether \liminf_{N→ ∞} ∏_r=1^N | 2sin πr α| =0 for all irrationals α. Despite its simple formulation, the question has remained unanswered. It was shown by Lubinsky in 1999 that the answer is yes if α has unbounded continued fraction coefficients, and it was suggested that the answer is yes in general. However, we show in this paper that for the golden ratio φ=(√(5)-1)/2, \liminf_{N→ ∞} ∏_r=1^N | 2sin πr φ| gt;0 , providing a negative answer to this long-standing open problem.

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