TY - JOUR
T1 - Ergodic averages for independent polynomials and applications
AU - Frantzikinakis, Nikos
AU - Kra, Bryna
N1 - Funding Information:
The first author was partially supported by NSF grant DMS-0111298 and the second author by NSF grant DMS-0244994.
PY - 2006/8
Y1 - 2006/8
N2 - Szemerédi's theorem states that a set of integers with positive upper density contains arbitrarily long arithmetic progressions. Bergelson and Leibman generalized this, showing that sets of integers with positive upper density contain arbitrarily long polynomial configurations; Szemerédi's theorem corresponds to the linear case of the polynomial theorem. We focus on the case farthest from the linear case, that of rationally independent polynomials. We derive results in ergodic theory and in combinatorics for rationally independent polynomials, showing that their behavior differs sharply from the general situation.
AB - Szemerédi's theorem states that a set of integers with positive upper density contains arbitrarily long arithmetic progressions. Bergelson and Leibman generalized this, showing that sets of integers with positive upper density contain arbitrarily long polynomial configurations; Szemerédi's theorem corresponds to the linear case of the polynomial theorem. We focus on the case farthest from the linear case, that of rationally independent polynomials. We derive results in ergodic theory and in combinatorics for rationally independent polynomials, showing that their behavior differs sharply from the general situation.
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U2 - 10.1112/S0024610706023374
DO - 10.1112/S0024610706023374
M3 - Review article
AN - SCOPUS:33747479524
VL - 74
SP - 131
EP - 142
JO - Journal of the London Mathematical Society
JF - Journal of the London Mathematical Society
SN - 0024-6107
IS - 1
ER -