TY - JOUR

T1 - Expansion of orbits of some dynamical systems over finite fields

AU - Gutierrez, Jaime

AU - Shparlinski, Igor E.

PY - 2010/10

Y1 - 2010/10

N2 - 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 〈→φ(〈) associated with a rational function φ ε 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 φ (X)=(aX+b)(cX+d)εFp(X), with ad≠ and c≠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.

AB - 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 〈→φ(〈) associated with a rational function φ ε 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 φ (X)=(aX+b)(cX+d)εFp(X), with ad≠ and c≠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.

UR - http://www.scopus.com/inward/record.url?scp=77957240709&partnerID=8YFLogxK

U2 - 10.1017/S0004972709001270

DO - 10.1017/S0004972709001270

M3 - Article

AN - SCOPUS:77957240709

SN - 0004-9727

VL - 82

SP - 232

EP - 239

JO - Bulletin of the Australian Mathematical Society

JF - Bulletin of the Australian Mathematical Society

IS - 2

ER -