On the general no-three-in-line problem

In this paper, we show that the number of points that can be placed in the grid $n\times n\times \cdots \times n(dtimes)=n^d$ for all $d\in \mathbb{N}$ with $d\ge 2$ so that no three points are collinear satisfies the lower bound \begin{align} \gg n^{d-1}\sqrt[2d]{d}.\nonumber \end{align} This extends the result of the no-three-in-line problem to all dimension $d\ge 3$.