当前位置: X-MOL 学术J. Number Theory › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
A proof of Sarnak's golden mean conjecture
Journal of Number Theory ( IF 0.6 ) Pub Date : 2020-09-01 , DOI: 10.1016/j.jnt.2020.02.005
C.J. Mozzochi

Abstract For an irrational number θ, let 0 = a 0 a 1 a 2 … a m a m + 1 = 1 be the sequence of points { l θ } , 1 ≤ l ≤ m (where { x } denotes x − [ x ] , the fractional part of x) and define d θ ( m ) = max ⁡ { ( a i − a i − 1 ) , 1 ≤ i ≤ m + 1 } Sarnak conjectured that sup ⁡ m d θ ⁎ m → ∞ ( m ) ≤ sup ⁡ m d θ m → ∞ ( m ) where θ ⁎ = 1 + 5 2 is the golden mean and θ is an arbitrary irrational number. In this paper we establish the conjecture, and we determine exactly sup ⁡ m d θ ⁎ ( m ) . Four special properties of the golden mean are crucial to the proof.

中文翻译:

萨纳克中庸猜想的证明

摘要 对于无理数 θ,设 0 = a 0 a 1 a 2 … amam + 1 = 1 为点序列 { l θ } , 1 ≤ l ≤ m(其中 { x} 表示 x − [ x ] , x) 的小数部分并定义 d θ ( m ) = max ⁡ { ( ai − ai − 1 ) , 1 ≤ i ≤ m + 1 } Sarnak 推测 sup ⁡ md θ ⁎ m → ∞ ( m ) ≤ sup ⁡ md θ m → ∞ ( m ) 其中 θ ⁎ = 1 + 5 2 是黄金平均值,θ 是任意无理数。在本文中,我们建立了猜想,并准确地确定了 sup ⁡ md θ ⁎ ( m ) 。中庸之道的四个特殊性质对证明至关重要。
更新日期:2020-09-01
down
wechat
bug