4 problems

Let $F$ be the limiting gap distribution function for the cases considered, and let $F'(s)$ denote its derivative. Conjecture. In the cases considered, $F'(s)$ exists, is continuou…

Let $\alpha>0$ be non-integer. Consider the fractional parts $\{n^\alpha\}$ for $n\in[N,2N)$, ordered increasingly, and normalize consecutive gaps at scale $1/N$. Folklore gap-dist…

Let $\alpha\neq 0$ and let $\theta>0$ satisfy $\theta\neq\tfrac12$ and $\theta\notin\mathbb Z$. For each $N$, let $y_1^{(N)}<\cdots<y_N^{(N)}$ be an ordering of the points…

Quadratic-residue gap second-moment conjecture. As $\omega(M)\to\infty$,