The distribution of spacings between the fractional parts of $\boldsymbol{n^dα}$

Sums of reciprocals of fractional parts

Let $\boldsymbolα\in \mathbb{R}^N$ and $Q\geq 1$. We consider the sum $\sum_{\boldsymbol{q}\in [-Q,Q]^N\cap\mathbb{Z}^N\backslash\{\boldsymbol{0}\}}\|\boldsymbolα\cdot\boldsymbol{q}\|^{-1}$. Sharp upper bounds are known when $N=1$, using continued fractions or the three distance theorem. However, these techniques do not seem to apply in higher dimension. We introduce a different approach, based on a general counting result of Widmer for weakly admissible lattices, to establish sharp upper bounds for arbitrary $N$. Our result also sheds light on a question raised by Lê and Vaaler in 2013 on the…

The distribution of spacings between the fractional parts of $n^2 α$

We study the distribution of normalized spacings between the fractional parts of an^2, n=1,2,.... We conjecture that if a is "badly approximable" by rationals, then the sequence of fractional parts has Poisson spacings, and give a number of results towards this conjecture. We also present an example of a Diophantine number a for which the higher correlation functions of the sequence blow up.

The distribution of spacings between fractional parts of lacunary sequences

Given an increasing sequence of integers a(n), it is known (due to Weyl) that for almost all reals t, the fractional parts of the dilated sequence t*a(n) are uniformly distributed in the unit interval. Some effort has been made recently to understand the distribution of the spacings between elements of such sequences (normalized to have unit mean), especially for sequences of polynomials. In this note, we consider lacunary sequences, such as the sequence of of powers of 2 (that is a(n)= 2^n). We show that for almost every t, all local correlation functions of the sequence t*2^n are those of a …

Gaps between logs

We calculate the limiting gap distribution for the fractional parts of log n, where n runs through all positive integers. By rescaling the sequence, the proof quickly reduces to an argument used by Barra and Gaspard in the context of level spacing statistics for quantum graphs. The key ingredient is Weyl equidistribution of irrational translations on multi-dimensional tori. Our results extend to logarithms with arbitrary base; we deduce explicit formulas when the base is transcendental or the r:th root of an integer. If the base is close to one, the gap distribution is close to the exponential…

多维量子谐振子的能级间距统计：代数情形

Level spacing statistics for the multi-dimensional quantum harmonic oscillator: algebraic case

针对频率比生成代数数域的多维量子谐振子，本文利用代数数论证明重标度能级间距、相邻间距比及有限间距词的分布均随 log E 渐近准周期。