On the distribution of $\hat{I}\pm p^2$ modulo one in the intersection of two Piatetski--Shapiro sets

Junyi Chu, Jinjiang Li, Min Zhang

Let $\lfloor t\rfloor$ denote the integer part of $t\in \mathbb{R}$ and $\Vert x\Vert$ the distance from $x$ to the nearest integer. Suppose that $1/2<{\hat{I}}_2^3<{\hat{I}}_1^3<1$ are two fixed constants.…