## Abstract

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‖U^{k}(G)≤η implies that A contains a non-trivial 3-term arithmetic progression with a common gap in S? Here, ‖{dot operator}‖U^{k}(G) denotes the kth Gowers norm.For G=Z_{n} we observe that k must be at least 3. However for G=F^{n}_{p} we show that k= 2 is sufficient, where here p is an odd prime and n is sufficiently large.

Original language | English |
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Pages (from-to) | 447-455 |

Number of pages | 9 |

Journal | Electronic Notes in Discrete Mathematics |

Volume | 49 |

DOIs | |

State | Published - Nov 2015 |

## Keywords

- Arithmetic progressions
- Pseudorandom sets
- Regularity method

## All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics
- Applied Mathematics