Counting integral points near a curve in the n-dimensional Euclidean space

We improve a result of Hickman and Srivastava, which gives upper and lower bounds for the number of integral points which lie within a neighborhood of a smooth non-degenerate curve in ${\mathbb{R}}^n$ for $n\ge 3$. Through more refined calculations, we have unleashed the great potential of the method in their work. Our improvement lies in the case $n\ge 4$, and the bounds are the best so far under certain conditions.