Kesten’s theorem holds for random Kronecker sequences on the torus T^d

You need to know: Matrix, determinant of a matrix, notation $AC$ for the image of set $C$ under linear transformation defined by matrix $A$ , notation $\times$ for direct product of sets, notation “ $x \mod 1$ ” for the real number $y\in[0,1)$ such that $x-y$ is an integer, notation ${\mathbb P}$ for the probability, selection uniformly at random.

Background: Let $d>0$ be an integer, $I=\{1,2,\dots,d\}$ , $T=[0,1)$ , and $T^d$ be set of vectors $x=(x_1, \dots, x_d)$ with $x_i \in T$ for every $i \in I$ . Let $U$ be the set of vectors $u=(u_1, \dots, u_d)$ such that $v_i\leq u_i \leq w_i$ for every $i \in I$ , where $v_i, w_i$ are fixed such that $0<v_i<w_i<1/2$ for every $i \in I$ . For each $u\in U$ , let $C_u$ be the set of vectors $y=(y_1, \dots, y_d)$ such that $|y_i|\leq u_i$ for every $i \in I$ . For a (small) $\eta>0$ , let $G_\eta$ be the set of $d\times d$ matrices with determinant $1$ and real entries $a_{ij}$ , such that $|a_{ii}-1|<\eta$ for all $i$ and $|a_{ij}|<\eta$ for all $i\neq j$ . Let $X=T^d \times T^d \times U \times G_\eta$ . For $\nu = (\alpha, x, u, A) \in X$ and integer $N>0$ , let $M(\nu, N)$ be the number of integers $1\leq m \leq N$ such that $(x+m\alpha) \mod 1 \in A C_u$ , and let $D(\nu, N)=M(\nu, N) - 2^d (\prod_{i=1}^d u_i) N$ .

The Theorem: On 19th November 2012 Dmitry Dolgopyat and Bassam Fayad submitted to arxiv a paper in which they proved that if $\nu$ is selected in $X$ uniformly at random, then, for all real $z$ , $\lim\limits_{N\to\infty}{\mathbb P}\left(\frac{D(\nu,N)}{(\ln N)^d}\leq z\right)=F(\rho_d z)$ , where $\rho_d$ is a constant depending only on $d$ , and $F(z)=\frac{\arctan(z)}{\pi}+\frac{1}{2}$ .

Short context: For every irrational $\alpha$ , it is known that sequence $\alpha, 2\alpha, \dots, m\alpha, \dots$ (called the Kronecker sequence) is uniformly distributed on $[0,1)$ , and the same is true for shifted sequence $x+\alpha, \dots, x+m\alpha, \dots$ . More formally, if $M(N)$ is the number of terms of this sequence (with $1\leq m\leq N$ ) belonging to some interval $[a,b) \subset [0,1)$ , then $\lim\limits_{N\to\infty}\frac{M(N)}{N}=b-a$ . Quantity $D(N)=M(N)-(b-a)N$ measures how fast this convergence happens. In 1962, Kesten established the limiting distribution of $D(N)$ , after appropriate scaling, provided that $x$ and $\alpha$ are selected at random. The Theorem establishes a multidimensional version of this result.