On the Distribution of Orbits of $\mathrm{PGL_2(q)}$ in ${\mathbbF}_{q^n}$ and the Klapper Conjecture
Igor E. Shparlinski · SIAM Journal on Discrete Mathematics · 2010
Motivated by a conjecture of Klapper [Finite Fields, Coding Theory, and Advances in Communications and Computing, Marcel Dekker, New York, 1993], we study the distribution of elements $\xi$ of a finite field $\mathbb{F}_{q^n}$ of $q^n$ elements under the action of the transformations $\xi\to(a\xi+b)/(c\xi+d)$ for matrices $\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)\in\mathrm{PGL_2(q)}$. We slightly improve a result of Niederreiter and Winterhof [Finite Fields Appl., 9 (2003), pp. 458–471] towards this conjecture. On the other hand, we also show that the original conjecture is false as stated.