On the Fractional Parts of Polynomials Modulo $p$

Abstract

We study a half-interval distribution problem for polynomial residues modulo an odd prime $p$: how often the fractional part of $\ddot{I}†(x)/p$ lies in the upper half of the unit interval as $x$ ranges over $1\le x<p/2$. Using finite Fourier expansions together with the Weil bound, we prove an asymptotic formula $\#\left\{1\le x<p/2:\left\{\ddot{I}†(x)/p\right\}>\frac{1}{2}\right\}=\frac{p}{4}+O_{\ddot{I}}†(\sqrt{p}{\log }^{2}p).$ We then show that the error term can be improved to $O_{\ddot{I}}†(\sqrt{p}\log p)$ for arbitrary quadratic polynomials and for polynomials satisfying suitable reflection symmetries. For even monomials $\ddot{I}†(x)=x^m$, we further obtain the bound $O_m(\sqrt{p}\log \log p)$ under the Generalized Riemann Hypothesis. Finally, in the case $m=2$, we prove an unconditional matching lower bound, showing that the factor $\log \log p$ is best possible in this setting.

12 pages