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On the distribution of rational functions on consecutive powers

Author(s): Jaime Gutierrez | Igor E. Shparlinski

Journal: Uniform Distribution Theory
ISSN 1336-913X

Volume: 3;
Issue: 1;
Start page: 85;
Date: 2008;
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Keywords: Rational functions | exponential sums

We show that for a prime $p$ and any nontrivialrational function $r(X) \in \F_p(X)$over the finite field $\F_p$ of $p$ elements, the fractionalparts$$ \left\{\frac{r(x)}{p}, \ldots, \frac{r(x^m)}{p} \right\},$$where $x$ runs through the fields elements which are not thepoles of the above functions,are asymptotically uniformly distributed in the $m$-dimensionalunit cube for any fixed $m$ and $p \to \infty$.
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