4 problems

For $\alpha\in[0,1]$, say that $\alpha$ is of type $\omega$ if there are only finitely many pairs $(a,q)$ satisfying $\lvert\alpha-a/q\rvert<q^{-\omega}$. Let $a/q$ range over the…

Let $Y$ be uniformly distributed on $[0,1)$, and for $N\in\mathbb{N}$ and $0<L\leq N$ define the random variable … Here $\operatorname{Po}(L)$ denotes a Poisson-distributed random…

Let $(\Lambda_j)$ be the locally normalized sequence of energy levels of an integrable Hamiltonian system, and let $r$ be fixed. For an interval $[X,X+r]$, write $k$ for the number…

Consider the one-dimensional Anderson model with independent site variables taking the two values $0$ and $1$, and with sufficiently small disorder. Let Proposition 4 denote the Po…