论文标题
余弦标志相关
Cosine Sign Correlation
论文作者
论文摘要
修复$ \ left \ {a_1，\ dots，a_n \ right \} \ subset \ mathbb {n} $，让$ x $为$ [0,2π] $上的均匀分布的随机变量。概率$ \ mathbb {p}（a_1，\ ldots，a_n）$ that $ \ cos（a_1 x），\ dots，\ cos（a_n x）$均为$零或全负为$零，因为$ \ cos（a_i x）\ sim x $ sim 1 $ for $ x $ $ x $ $ x $ $ 0 $ 0 $ 0 $ 0。我们对这种概率的较小感兴趣。由光谱理论中的问题激励，冈卡维尔斯，奥利维拉·席尔瓦和斯坦纳伯格证明了$ \ mathbb {p}（a_1，a_1，a_1，a_2）\ geq 1/3 $等于且仅在$ \ left \ left \ left \ left \ left \ e_1，a_2，a_2 \ right \ right \ right \ right \ right \ right \ phcd = gcd（a_1）时3 \ right \} $。 We prove $\mathbb{P}(a_1,a_2,a_3)\geq 1/9$ with equality if and only if $\left\{a_1, a_2, a_3 \right\} = \gcd(a_1, a_2, a_3)\cdot \left\{1, 3, 9\right\}$.该模式不会继续，因为$ \ left \ {1,3,11,33 \ right \} $达到的值小于$ \ left \ {1,3,9,27 \ right \} $。我们推测$ \ left \ {1,3,11,33 \ right \} $的倍数是$ n = 4 $的最佳倍数，讨论对Schrödingeroperators $-Δ+ V $的特征的含义，并以孤独的跑步者的问题对问题进行解释。
Fix $\left\{a_1, \dots, a_n \right\} \subset \mathbb{N}$, and let $x$ be a uniformly distributed random variable on $[0,2π]$. The probability $\mathbb{P}(a_1,\ldots,a_n)$ that $\cos(a_1 x), \dots, \cos(a_n x)$ are either all positive or all negative is non-zero since $\cos(a_i x) \sim 1$ for $x$ in a neighborhood of $0$. We are interested in how small this probability can be. Motivated by a problem in spectral theory, Goncalves, Oliveira e Silva, and Steinerberger proved that $\mathbb{P}(a_1,a_2) \geq 1/3$ with equality if and only if $\left\{a_1, a_2 \right\} = \gcd(a_1, a_2)\cdot \left\{1, 3\right\}$. We prove $\mathbb{P}(a_1,a_2,a_3)\geq 1/9$ with equality if and only if $\left\{a_1, a_2, a_3 \right\} = \gcd(a_1, a_2, a_3)\cdot \left\{1, 3, 9\right\}$. The pattern does not continue, as $\left\{1,3,11,33\right\}$ achieves a smaller value than $\left\{1,3,9,27\right\}$. We conjecture multiples of $\left\{1,3,11,33\right\}$ to be optimal for $n=4$, discuss implications for eigenfunctions of Schrödinger operators $-Δ+ V$, and give an interpretation of the problem in terms of the lonely runner problem.
