The distribution of values of short hybrid exponential sums on curves over finite fields

Let $p$ be a prime number, $X$ be an absolutely irreducible affine plane curve over ${\mathbb{F}}_p$, and $g,f\in {\mathbb{F}}_p(x,y)$. We study the distribution of the values of the hybrid exponential sums S_n on $n\in \mathcal{I}$ for some short interval $\mathcal{I}$. We show that under some natural conditions the limiting distribution of the projections of the sum $S_n$, $n\in \mathcal{I}$ on any straight line through the origin is Gaussian as $p$ tends to infinity.