Counting integral points near space curves: a Fourier analytic approach

Abstract

We establish upper and lower bounds for the number of integral points which lie within a neighbourhood of a smooth nondegenerate curve in $\mathbb{R}^n$ for $n\geq 3$. These estimates are new for $n\geq 4$, and we recover an earlier result of J. J. Huang for $n=3$. However, we do so by using Fourier analytic techniques which, in contrast with the method of Huang, do not require the sharp counting result for planar curves as an input. In particular, we rely on an Arkhipov--Chubarikov--Karatsuba-type oscillatory integral estimate.