On the distribution of the \({\mathbb{F}}_p\)-points on an affine curve in \(r\) dimensions (Q2759136)

Let \(C\) be a geometrically irreducible degree-\(d\) curve in affine \(r\)-space \({\mathbb A}^r\) over a finite field~\({\mathbb F}_p\), and assume that \(C\) is not contained in any hyperplane. It is natural to expect that the \({\mathbb F}_p\)-rational points of \(C\) should be distributed somewhat randomly throughout~\({\mathbb A}^r({\mathbb F}_p)\). The author formulates and proves a precise version of this statement. NEWLINENEWLINENEWLINELet \({\mathbb T}^r\) be the torus \(({\mathbb R}/{\mathbb Z})^r\), and choose an embedding \(t\) of the additive group \({\mathbb A}^r({\mathbb F}_p)\) into~\({\mathbb T}^r\). Let \(\Omega\) be a domain in \({\mathbb T}^r\) with piecewise smooth boundary. The author shows that the fraction of points \(P\) of \(C\) with \(t(P) \in\Omega\) differs from the volume of \(\Omega\) by at most NEWLINE\[NEWLINEc_{r,d,\Omega} p^{-1/(2r+2)} \log^{r/(r+1)} p,NEWLINE\]NEWLINE where \(c_{r,d,\Omega}\) is a constant that does not depend on the prime~\(p\). The main ingredient of the proof is the Bombieri-Weil inequality [\textit{E. Bombieri}, Am. J. Math. 88, 71-105 (1966; Zbl 0171.41504)]. NEWLINENEWLINENEWLINEThe results of this paper generalize results of \textit{Z. Zheng} [J. Number Theory 59, 106-118 (1996; Zbl 0862.11041)]. Zheng considered the case \(r=2\), that is, curves in the affine plane. Zheng stated his result only for one particular family of domains \(\Omega\subset{\mathbb T}^2\), but his argument applied to more general domains as well.