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：

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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。