5 problems

For a real number $\alpha$, consider the sequence $(\{n^2\alpha\})_{n\ge1}$, where $\{x\}$ denotes the fractional part of $x$. Its pair correlation is Poissonian when it agrees wit…

Let $b$ and $c$ be non-zero integers such that $3\nmid bc$. For a sequence … let $D_{\operatorname{qr}}(n)$ denote its discriminator, the least modulus for which the first $n$ sequ…

For a real number $\alpha$, consider the sequence $(\{n^2\alpha\})_{n\ge1}$, where $\{x\}$ denotes the fractional part of $x$. Its triple and higher correlations are called Poisson…

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…