Poisson spacing statistics for value sets of polynomials

Poisson spacing statistics for lattice points on circles

We show that along a density one subsequence of admissible radii, the nearest neighbor spacing between lattice points on circles is Poissonian.

Poisson statistics via the Chinese remainder theorem

We consider the distribution of spacings between consecutive elements in subsets of Z/qZ where q is highly composite and the subsets are defined via the Chinese remainder theorem. We give a sufficient criterion for the spacing distribution to be Poissonian as the number of prime factors of q tends to infinity, and as an application we show that the value set of a generic polynomial modulo q have Poisson spacings. We also study the spacings of subsets of Z/q_1q_2Z that are created via the Chinese remainder theorem from subsets of Z/q_1Z and Z/q_2Z (for q_1,q_2 coprime), and give criteria for wh…

Spacings between integers having typically many prime factors

We show that the sequence of integers which have nearly the typical number of distinct prime factors forms a Poisson process. More precisely, for $\de$ arbitrarily small and positive, the nearest neighbor spacings between integers $n$ with $|\om(n)-\log_2 n|\le (\log_2 n)^{\de}$ obey the Poisson distribution law.

平方和间隙的泊松分布与环形点散射器的能级间距

Poisson distribution for gaps between sums of two squares and level spacings for toral point scatterers

本文研究了平方和间隙的泊松分布，并证明了环形点散射器的能级间距在弱耦合极限下也服从泊松分布。

Nearest neighbor spacing distribution of prime numbers and quantum chaos

We give heuristic arguments and computer results to support the hypothesis that, after appropriate rescaling, the statistics of spacings between adjacent prime numbers follows the Poisson distribution. The scaling transformation removes the oscillations in the NNSD of primes. These oscillations have the very profound period of length six. We also calculate the spectral rigidity $Δ_3$ for prime numbers by two methods. After suitable averaging one of these methods gives the Poisson dependence $Δ_3(L)=L/15$.

立方三维环面上点散射体的间距统计

Spacing statistics for a point scatterer on the cubic three-torus

研究立方三维环面上点散射体的新特征值间距统计，证明其收敛于独立于参数的分布，并刻画了小间隙渐近形式。