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 \geq 3 $. Through more refined calculations, we have unleashed the great potential of the method in their work. Our improvement lies in the case $ n \geq 4 $, and the bounds are the best so far under certain conditions.