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I am interested in knowing about the distribution of exponentially high order elements in $\mathbb{F}_p$. To be precise let $s$ be of the order $\frac{p}{\log^{k}(p)}$ for some fixed $k$ and integer. Given $p$ being an odd prime. I would like to know about the distribution of elements with order higher than $s$. Are there papers dealing with such a question? I could not find any resources regarding this.

Thanks!!

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  • $\begingroup$ They equidistribute in $\mathbb{F}_p$ as $p$ tends to infinity. This amounts to control the Gauss sums $\sum_{x \in \mathbb{F}_p} \exp{\frac{2 i \pi \lambda x^{\alpha}}{p}}$, with $\alpha \leq \frac{p-1}{s}$ dividing $p-1$, which is standard. $\endgroup$
    – js21
    Commented May 18, 2017 at 12:10

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A paper by Bourgain will help. Here is the link.

Bourgain's result is:

Let $p$ be a prime and $H$ be a subgroup of the multiplicative group $\mathbb{Z}_p^*$. There exists positive constants $C_1>1$ and $C$ such that if $|H|> p^{C_1/\log\log p}$ then $$ \max_{(k,p)=1} \left| \sum_{a\in H} \exp\left(2\pi i k a /p \right)\right|=O\left( \exp(-(\log p)^C) |H| \right).$$

I also wrote a paper applying Bourgain' result to obtain an equidistribution result. This is Corollary 2.6 of my paper.

The result is:

Let $p$ be a prime and $y\geq 1$. Let $d|p-1$ and $d> p^{C_1/\log\log p}$ for some $C_1>1$. Then there exists positive constant $C$ such that $$ |a<y: a^d\equiv 1 (p)|= \frac {yd}p +O(d\exp(-(\log p)^C)). $$

The method is a standard method that is frequently used, which links equidistribution result with exponential sum bound by using Erdos Turan inequality.

However, this result is equidistribution of the elements generated by an element of high order, not the elements of high orders.

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