TY - JOUR

T1 - Arithmetic progressions with a pseudorandom step

AU - Aigner-Horev, Elad

AU - Hàn, Hiêp

N1 - Publisher Copyright:
© 2015 Elsevier B.V.

PY - 2015/11

Y1 - 2015/11

N2 - Let α, σ > 0 and let A and S be subsets of a finite abelian group G of densities α and σ, respectively, both independent of |G|. Without any additional restrictions, the set A need not contain a 3-term arithmetic progression whose common gap lies in S. What is then the weakest pseudorandomness assumption that if put on S would imply that A contains such a pattern?More precisely, what is the least integer k≥2 for which there exists an η=η(α, σ) such that ‖S-σ‖Uk(G)≤η implies that A contains a non-trivial 3-term arithmetic progression with a common gap in S? Here, ‖{dot operator}‖Uk(G) denotes the kth Gowers norm.For G=Zn we observe that k must be at least 3. However for G=Fnp we show that k= 2 is sufficient, where here p is an odd prime and n is sufficiently large.

AB - Let α, σ > 0 and let A and S be subsets of a finite abelian group G of densities α and σ, respectively, both independent of |G|. Without any additional restrictions, the set A need not contain a 3-term arithmetic progression whose common gap lies in S. What is then the weakest pseudorandomness assumption that if put on S would imply that A contains such a pattern?More precisely, what is the least integer k≥2 for which there exists an η=η(α, σ) such that ‖S-σ‖Uk(G)≤η implies that A contains a non-trivial 3-term arithmetic progression with a common gap in S? Here, ‖{dot operator}‖Uk(G) denotes the kth Gowers norm.For G=Zn we observe that k must be at least 3. However for G=Fnp we show that k= 2 is sufficient, where here p is an odd prime and n is sufficiently large.

KW - Arithmetic progressions

KW - Pseudorandom sets

KW - Regularity method

UR - http://www.scopus.com/inward/record.url?scp=84947763963&partnerID=8YFLogxK

U2 - 10.1016/j.endm.2015.06.063

DO - 10.1016/j.endm.2015.06.063

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:84947763963

SN - 1571-0653

VL - 49

SP - 447

EP - 455

JO - Electronic Notes in Discrete Mathematics

JF - Electronic Notes in Discrete Mathematics

ER -