Abstract: Given a finite field Fp = {0, . . . , p-1} of p elements, where p is a prime, we consider the distribution
of elements in the orbits of a transformation ? 7? ?(? ) associated with a rational function ? ? Fp(X).
We use bounds of exponential sums to show that if N ? p1/2+? for some fixed ? then no N distinct
consecutive elements of such an orbit are contained in any short interval, improving the trivial lower
bound N on the length of such intervals. In the case of linear fractional functions ?(X) = (a X + b)/(cX + d) ? Fp(X), with ad 6= bc and c 6= 0, we use a different approach, based on some results of additive combinatorics due to Bourgain, that gives a nontrivial lower bound for essentially any admissible value of N
Authorship: Gutierrez J., Shparlinski I.,
Fuente: Bulletin of the Australian Mathematical Society, 2010, 82(2), 232-239
Publisher: Cambridge University Press
Publication date: 01/10/2010
No. of pages: 8
Publication type: Article
DOI: 10.1017/S0004972709001270
ISSN: 0004-9727,1755-1633
Spanish project: MTM2007-67088
Publication Url: https://doi.org/10.1017/S0004972709001270